Paul Meany rightly points out the dependence of the classical liberal tradition generally, and the American founders in particular, on thinkers of classical antiquity such as Cicero. But the story is more complicated, the relationship a complex mixture of appreciation and rejection.
Thomas Jefferson, for example, admired the ancients, and even called himself an Epicurean. But he also wrote:
Let a just view be taken of the moral principles inculcated by the most esteemed of the sects of antt. [= ancient] philosophy, or of their individuals; particularly Pythagoras, Socrates, Epicurus, Cicero, Epictetus, Seneca, Antoninus [= Marcus Aurelius] ….
1. Their precepts related chiefly to ourselves, and the government of those passions which, unrestrained, would disturb our tranquility of mind. In this branch of Philosophy they were really great. …
2. In developing our duties to others, they were short and defective. They embraced indeed the circles of kindred & friends; and inculcated patriotism, or the love of our country in the aggregate, as a primary obligation: towards our neighbors & countrymen, they taught justice, but scarcely viewed them as within the circle of benevolence. Still less have they inculcated peace, charity, & love to our fellow men, or embraced, with benevolence, the whole family of mankind.
By contrast, Jefferson credits Jesus—whom he regards more as a moral philosopher and teacher than as any sort of supernatural figure—with improving on the Greeks and Romans by, inter alia, “inculcating universal philanthropy, not only to kindred and friends, to neighbors and countrymen, but to all mankind, gathering all into one family, under the bonds of love, charity, peace, common wants, and common aids.” Or as he put it elsewhere: “Epictetus & Epicurus give us laws for governing ourselves, Jesus a supplement of the duties & charities we owe to others.” (J. S. Mill and Harriet Taylor, in On Liberty, would similarly call for a balance between “pagan self-assertion” and “Christian self-denial.”)
This might seem unfair to thinkers like Cicero, who—drawing on the Stoic idea of the cosmopolis—explicitly taught the universality of the duties of justice and benevolence throughout the community of humankind. And in his De Officiis, Cicero even says that “there are two ways of contending, one by discussion, the other by force, the former belonging properly to man, the latter to beasts,” and that the way of force must be chosen only when the way of persuasion is not available—about as libertarian a thesis as one could desire.
But however liberal and universalist Cicero’s principles may sound, given his generally approving attitude toward Roman foreign policy throughout the republican period, including the capture of both territory and slaves through conquest, Cicero can fairly be charged with failing to apply those principles in practice. And Cicero’s vision of the political order as properly and primarily devoted to the protection of property rights arguably loses whatever lustre it might have had once we recall that the property to be protected consisted largely of stolen land and enslaved human beings. One can fairly see these failures of Cicero, and of the regime he championed, as rooted in an inadequate recognition of the equal claims of those outside the Roman elite to be objects of moral concern, however eloquently Cicero may have made lip service to moral universalism. (Though on the other hand, neither Jesus nor the early Christians condemn imperialism or slavery in any straightforward and unambiguous way either. And in any case, Jefferson himself—eloquent defender of liberty in theory, yet, in practice, both slaveowner and trans-continental expansionist—is perhaps in no strong position to criticize Cicero here.)
Later in the century, many admirers of the American republic and individual liberty took a still more jaundiced view of classical antiquity, though in a way that developed some of the concerns that worried Jefferson. Benjamin Constant, in his celebrated 1819 essay The Liberty of the Ancients Compared with That of the Moderns (1819), contrasted the “modern” conception of liberty, understood as including
the right of everyone to express their opinion, choose a profession and practice it, to dispose of property, and even to abuse it; to come and go without permission, and without having to account for their motives or undertakings …. to associate with other individuals, either to discuss their interests, or to profess the religion which they and their associates prefer, or even simply to occupy their days or hours in a way which is most compatible with their inclinations or whims ….
with the “ancient” conception of liberty, which Constant saw as a matter of a purely “collective freedom” entailing “the complete subjection of the individual to the authority of the community” (while making a partial exception for Athens—but not for Rome, even republican Rome):
All private actions were submitted to a severe surveillance. No importance was given to individual independence, neither in relation to opinions, nor to labor, nor, above all, to religion. The right to choose one’s own religious affiliation, a right which we regard as one of the most precious, would have seemed to the ancients a crime and a sacrilege. In the domains which seem to us the most useful, the authority of the social body interposed itself and obstructed the will of individuals. Among the Spartans, Therpandrus [i.e. Terpander] could not add a string to his lyre without causing offense to the ephors. In the most domestic of relations the public authority again intervened. The young Lacedaemonian could not visit his new bride freely. In Rome, the censors cast a searching eye over family life. The laws regulated customs, and as customs touch on everything, there was hardly anything that the laws did not regulate.
(I shall incidentally be arguing that Constant’s preference for democratic Athens over republican Rome is actually more defensible than the reverse preference held by most of the American founders.)
The classical liberal economist Frédéric Bastiat was still more hostile to the Roman legacy in particular, pillorying Rome in his 1850 “Academic Degrees and Socialism” as
a nation that, because it hates and despises work, has based its entire means of existence on the successive pillage of all the neighboring tribes and on slavery. This nation has established a policy, a moral code, a religion, and public opinion in line with the cruel purpose that is sustaining and developing it. …
We, for our part, say: “Men are the owners of themselves and consequently of their faculties and, following this, of the products of their faculties.” But could the Romans conceive of such a notion? As the owners of slaves, were they able to say: “Man belongs to himself”? As they despised work, were they able to say: “Man is the owner of the product of his faculties”?
Bastiat accordingly laments the tendency of traditional education to, in effect,
send all French young people to visit this [Roman] nation, to live its way of life, be inspired by its sentiments, share its enthusiasms, and breathe its ideas as their own air …. to become imbued and permeated with and saturated in the sentiments and opinions of a nation of brigands and slaves …. 
and Bastiat sees the popularity of militarism and collectivism as the inevitable result of such classical education.
From my own perspective, Constant’s and Bastiat’s condemnations of classical antiquity are accurate as far as they go, but one-sided; and the admiring vision that many of the American founders held of (republican) Rome and of Cicero is also accurate as far as it goes, but again one-sided. Sifting the wheat from the tares is a delicate operation.
I’ve argued elsewhere that anticipations of libertarian and classical liberal themes are to be found as deep-seated threads throughout both the ideas and the practice of classical antiquity. For example, one can even see a proto-Hayekian moment in Cicero when he writes, in his Republic:
[Cato] used to say that the government of Rome was superior to that of other states; because in them the great men were mere isolated individuals, who regulated their constitutions according to their own ipse dixits, their own laws, and their own ordinances. … Our Roman constitution, on the contrary, did not spring from the genius of an individual, but of many; and it was established, not in the lifetime of a man, but in the course of ages and centuries. For … there never yet existed a genius so vast and comprehensive as to allow nothing to escape its attention, and all the geniuses in the world united in a single mind, could never, within the limits of a single life, exert a foresight sufficiently extensive to embrace and harmonize all, without the aid of experience and practice.
Or again, Aristotle’s claim that we cherish the products of our labor because “the work is, in a way, the producer in activity” can be seen as a forerunner of Locke’s labor-mixing theory of property (an element that Meany rightly notes as missing in Cicero’s account).
But the extent to which themes of a tendency precisely the opposite of liberal or libertarian are likewise to be found in the ideas and practice of classical antiquity should not be lost sight of. This is a context in which sweeping generalizations in either direction are perilous.
The American Founders and Athenian Democracy
As Meany notes, the American Founders tended to reserve their admiration for republican Rome and not for democratic Athens, dismissing the latter as “little better than mob rule,” and thus “not a blueprint for a political model but instead a cautionary tale of unfettered democracy” (unlike Constant, who, as we’ve seen, looked more favorably on Athens than on Rome, despite regarding the two as all too similar).
James Madison, for example, charges in Federalist 10 that “a pure democracy … can admit of no cure for the mischiefs of faction,” and that “such democracies have ever been spectacles of turbulence and contention; have ever been found incompatible with personal security, or the rights of property; and have, in general, been as short in their lives, as they have been violent in their deaths.” In Federalist 55, Madison adds that “[h]ad every Athenian citizen been a Socrates, every Athenian assembly would still have been a mob.”
In their breezy dismissal of Athenian democracy as inferior to republican Rome, the American founders arguably show excessive servility to the caricatures perpetrated by anti-democratic writers like Plato, Aristotle, and Thucydides. Democratic Athens, far from being a site of mob rule, featured a complex system of legal checks and balances designed to prevent abuse of power by either minorities or majorities (including judicial review of measures passed by the democratic assembly); and while far from satisfying contemporary liberal or libertarian criteria of liberty even for its free adult male citizens—let alone for, e.g., women and slaves—Athens seems to have provided a level of both economic freedom and freedom of expression that had few rivals in the ancient world.
As for being short in life and violent in death, the Athenian democratic system in its classical form lasted, roughly, from 507-338 BCE—thus, a century and three quarters—with short-lived interruptions in 411 and 404-3; and its final overthrow was due to Macedonian imperialism (which saw the downfall of all the Greek city-states, of whatever constitutional form) more than to domestic infighting. Given that the American republic was to survive only a century or so before being hit by a devastating civil war, it’s not clear that Athens is such an inferior model in comparison with the United States.
The American Founders and Ancient Anarchism
Some classical thinkers were inclined toward views that would nowadays be identified as anarchist. For example, Zeno of Citium, founder of Stoicism, set out a vision of an ideal community of the wise, without law courts, government coinage, or prescribed gender roles; while the Epicurean Diogenes of Oenoanda looked forward to an era when the world will be filled with friendship, and there will be no further need for “walls or laws and all the things we contrive on account of one another.” Even Plato, in the second book of his Republic, suggests that his ideal state is only second-best to the anarchistic “city of pigs.”
One feature of these anarchistic utopias is that they are designed only for the wise and virtuous, which might suggest that government is necessary for social order so long as the majority of people are morally and intellectually imperfect. But Protagoras (in the Platonic dialogue of that name) argues that such social order as we enjoy would be impossible unless the average person possessed an adequate minimum of wisdom and virtue, while Glaucon (in the second book of Plato’s Republic) maintains that the need to retain a reputation as a cooperator is enough to make social cooperation in most people’s interest, most of the time. Both Protagoras and Glaucon portray the reinforcing knowledge (Protagoras) or incentives (Glaucon) for cooperation primarily in terms of horizontal inputs from one’s peers than vertical sanctions from state authority; thus their version of social cooperation opens the door to anarchism, even if they don’t walk through it. (And Aristophanes, in his play Acharnians presents—albeit with tongue in cheek—the attractions of a quasi-anarchistic policy of individual secession, with private citizens being able to formulate their own foreign policy in matters of war and trade, a policy of a more peaceful sort than that preferred by the Athenian state.)
Thomas Paine, whether consciously or not, follows the lead of some of these anarchistic or quasi-anarchistic ancient thinkers when he declares, in his Rights of Man:
Great part of that order which reigns among mankind is not the effect of government. It has its origin in the principles of society and the natural constitution of man. It existed prior to government, and would exist if the formality of government was abolished. … In fine society performs for itself almost everything which is ascribed to government. … Government is no farther necessary than to supply the few cases to which society and civilisation are not conveniently competent; and instances are not wanting to show, that everything which government can usefully add thereto, has been performed by the common consent of society, without government. … Formal government makes but a small part of civilised life …. The more perfect civilisation is, the less occasion has it for government, because the more does it regulate its own affairs, and govern itself ….
In explaining what makes social order possible, Paine appeals, like Glaucon, to self-interest: “The mutual dependence and reciprocal interest which man has upon man, and all the parts of civilised community upon each other, create that great chain of connection which holds it together. … Common interest regulates their concerns ….” But like Protagoras, he also appeals to widespread moral propensities that go beyond merely strategic concerns: “[Nature] has not only forced man into society by a diversity of wants which the reciprocal aid of each other can supply, but she has implanted in him a system of social affections, which, though not necessary to his existence, are essential to his happiness.”
What led Paine to pass from declaring government a “necessary evil” in his 1776 Common Sense, to downgrading it as unnecessary, or only barely necessary, 16 years later in his 1792 Rights of Man? Paine himself points to his own experience during the Revolutionary War:
For upwards of two years from the commencement of the American War, and to a longer period in several of the American States, there were no established forms of government. The old governments had been abolished, and the country was too much occupied in defence to employ its attention in establishing new governments; yet during this interval order and harmony were preserved as inviolate as in any country in Europe.
Ironically enough, Paine’s soon-to-be-archenemy Edmund Burke had made precisely the same observation in his 1775 speech on Conciliation with the Colonies. Another possible influence is that of Adam Smith’s demonstration of the spontaneously self-regulating nature of economic order in his Wealth of Nations (1776), a work which Paine admired.
Nor was Paine the only founder with anarchistic sympathies. Alexander Hamilton famously accused Aaron Burr of preaching “perfect Godwinism,” a reference to the theories of the English anarchist William Godwin (himself a friend of Paine); and while Hamilton may not be the most reliable of sources on the topic of Burr’s views, it’s undisputed that Burr was a friend and admirer of William Godwin and his family. Jefferson, moreover, acknowledged the temptations of anarchism in a letter to Madison:
Societies exist under three forms sufficiently distinguishable. 1. Without government, as among our Indians. 2. Under governments wherein the will of every one has a just influence, as is the case in England in a slight degree, and in our states in a great one. 3. Under governments of force: as is the case in all other monarchies and in most of the other republics. … It is a problem, not clear in my mind, that the 1st. condition is not the best. But I believe it to be inconsistent with any great degree of population.
To the extent that anarchism has its roots in Greco-Roman antiquity, it’s clear that the founders’ inheritance of the classical legacy was not confined to Roman-style republicanism.
 “I too am an Epicurean. I consider the genuine (not the imputed) doctrines of Epicurus as containing every thing rational in moral philosophy which Greece & Rome have left us.” (Thomas Jefferson to William Short, 31 Oct 1819.
 Jefferson, Doctrines of Jesus Compared With Others (21 Apr 1803).
 Jefferson, Doctrines of Jesus, op. cit.
 Jefferson to William Short, op. cit.
 John Stuart Mill and Harriet Taylor Mill, On Liberty III. (J. S. Mill identified Taylor as a co-author of several of the works published under his sole name, including On Liberty; I take there to be a presumption in favour of accepting his claim in the absence of adequate evidence to the contrary.)
 Cicero, On Moral Duties I.11.
 Benjamin Constant, The Liberty of the Ancients Compared With That of the Moderns (1819).
 Frédéric Bastiat, “Baccalaureate and Socialism” (1850).
 I explore proto-liberal, proto-libertarian, and proto-anarchistic themes in Greek and Roman philosophy in the following works:
“The Classical Roots of Radical Individualism” (Social Philosophy and Policy 24, no. 2 (Summer 2007), pp. 262-297; reprinted in David Keyt and Fred D. Miller, Jr., eds., Freedom, Reason, and the Polis: Essays in Ancient Greek Political Philosophy (Cambridge 2007), pp. 262-297);
“Socrates and Early Socratic Philosophers of Law” (with R. F Stalley; in Fred D. Miller, Jr., and Carrie-Ann Biondi, eds., A Treatise of Legal Philosophy and General Jurisprudence, Volume 6: A History of the Philosophy of Law from the Ancient Greeks to the Scholastics (Dordrecht: Springer, 2007), pp. 35-56);
“Aristotle’s Conception of Freedom” (Review of Metaphysics 49 (June 1996), pp. 775-802);
“Aristotle’s Egalitarian Utopia: The Polis kat’ euchen” (in Mogens Herman Hansen, ed. The Imaginary Polis: Symposium, January 7-10, 2004. Acts of the Copenhagen Polis Centre 7 (Copenhagen: Det Kongelige Danske Videnskabernes Selskab, 2005), pp. 164-196);
I also discuss proto-libertarian ideas in the Greek poets and playwrights in my Libertarianism.org series “Ancient Greece’s Legacy for Liberty” (27 parts at present writing, though I hope to continue it in the future), beginning here; see especially Parts 1-6 and 12-27.
 Cicero, On the Commonwealth II.
 Aristotle, Nicomachean Ethics 1167b31-1168a15.
 James Madison, Federalist §10.
 Madison, Federalist §55.
 I discuss the libertarian aspects of the Athenian legal system in:
“The Athenian Constitution: Government by Jury and ReferendumFormulations IV.1 (Autumn 1996), pp. 7-18, 21-23, 35);
“Civil Society in Ancient Greece: The Case of Athens” (Conference on Civil Society, Liberty Fund, Arlington VA, 29 May 1998);
and in Parts 7-11 of my “Ancient Greece’s Legacy for Liberty” series (op. cit.), beginning here
(See also chapter 16, “Athenian Law: The Work of a Mad Economist,” of David Friedman’s book Legal Systems very Different From Ours (2019). A draft of the chapter is available on Friedman’s website here.)
 Long, “Hellenistic Philosphers of Law,” op. cit. Note that this Zeno and Diogenes are not to be confused with the better-known Zeno of Elea (famous for his paradoxes of motion) or Diogenes of Sinope (the barrel-dwelling Cynic).
 Thomas Paine, Rights of Man, II.1 (1792).
 “Society in every state is a blessing, but Government, even in its best state, is but a necessary evil; in its worst state an intolerable one ….” Common Sense (1776).
 Paine, Rights of Man II.1, op. cit.
 “Pursuing the same plan of punishing by the denial of the exercise of Government to still greater lengths, we wholly abrogated the ancient Government of Massachusetts. We were confident that the first feeling, if not the very prospect of anarchy, would instantly enforce a complete submission. The experiment was tried. A new, strange, unexpected face of things appeared. Anarchy is found tolerable. A vast province has now subsisted, and subsisted in a considerable degree of health and vigour, for near a twelvemonth, without Governor, without public council, without judges, without executive magistrates. How long it will continue in this state, or what may arise out of this unheard-of situation, how can the wisest of us conjecture?” Edmund Burke, For Conciliation with the Colonies (1775).
 Bart Schultz, The Happiness Philosophers: The Lives and Works of the Great Utilitarians (Princeton U. P., 2017), pp. 38-39.