When did people start saying about the US that "it's a free country"?

Allthough quoting a Hollywood film (The Patriot, 2000) can never be considered a suitable source:

It's a free country. Or at least it will be.

probably expresses a realistic sediment of what the peaple of that time felt.

Free of domination from the outside, to deal with their own affairs.

The journals of the day probably used some (future tense) form of 'free country' to express what they wanted to achieve and after achieving that goal the present tense would have been used.

Searching through journals after July 1776, the term would probably be found.

The saying seems to predate 1826, when Edward Thornton Tayloe wrote (in Mexico 1825-1828, p. 128) that:

We have more than once since we have been in Mexico been induced to inquire if we were in a free country.

Not sure if you are asking about the complete expression 'it's a free country', but the concept of freedom and the shorter expression 'free country' was already used when the USA was created. In general, if you read Burke, it is always there.

Burke generally liked America, he was against the war, against the taxation that broke the colonial pact, and before the war he (not officially, I think) represented their interests in Parliament - remember colonies did not have MPs at London. So Burke knew american politicians (at least by letters) and the founders certainly would read Burke. If the expression was already common place in Burke, then there is nothing new about using it about the USA since the beginning.

This link has some Burke quotes on freedom. How many of them could an American use today, or last century?

And here 2 uses of the expression 'free country' by Burke:

In a free country every man thinks he has a concern in all public matters,--that he has a right to form and a right to deliver an opinion on them. This it is that fills countries with men of ability in all stations.

or (link):

is but a part of a rich commercial nation, the interests of which are various, multiform, and intricate. We are members for that great nation, which, however, is itself but part of a great empire, extended by our virtue and our fortune to the farthest limits of the East and of the West. All these wide-spread interests must be considered--must be compared--must be reconciled, if possible. We are members for a free country; and surely we all know that the machine of a free constitution is no simple thing, but as intricate and as delicate as it is valuable. We are members in a great and ancient monarchy; and we must preserve religiously the true legal rights of the sovereign, which form the keystone that binds together the noble and well-constructed arch of our empire and our Constitution. A constitution made up of balanced powers must ever be a critical thing.

Burke would see the English Constitution as a balance between democracy, monarchy, and aristocracy, not wholly different as the Americans would see the US fed government as a balance between democracy, federalism, and the rule of law. Individual rights and limited government might be traced in England to the Magna Carta, and in general to the christian concept of the individual as an image of god, or to greek philosophers. The idea of a free country was not a new thing under the sun.

As far as I understand this question, it seems to be based on false assumptions.

When did Americans start to use the expression?

Immediately when they became Americans? In fact, when they were still loyal subjects of the king! As they also were lucky enough to have had the Rights of Englishman. It was inherited?

Longman dictionary:

it’s a free country
British English

The Oxford Lexico uses "it's a free country" even in this example sentence for 'Englishman':

‘The message is: it's a free country, and an Englishman's home is his castle - just as long as you don't happen to live under the ridiculous rules of a Residents' Association!’

As such the meme occurs in 1748: James Digges La Touche: "A Free-Man's Answer to the Free-Holder's Address [signed: Hibernicus, i.e. J. D. Latouche] to the Merchants, Traders, and others ... of the City of Dublin. [Signed: Britannicus.]" (p3), and countless earlier examples.

And it seems to be not even a special 'English' thing, as an English translation from Italian in 1656/1669 uses 'the meme' as well:

… we cleerly see by this your misèrable spectacle, that nothing is sweeter, that there is no greater consolation, no greater Jubile of content then by forgetting injuries, pardoning offences, and embracing enemies, to live in a free Country, in such …

The earliest example on Google is dated 1607, but being a snippet view only and looking at the print, that seems to be a mistake.

Although my guess is that the underlying notion was used in Roman and even classical Greek times, think versus Persians…

Actually using it in its idiomatic form and meaning not in theoretical texts but in speech may be much younger though:

‘US, since before 1930s’, writes R.C., 1978, of it’s a free country, ‘first as an expression of tolerance, but, since [during the] 1950s, sometimes ironic, since it has become apparent that we are not quite so free as we thought’. P.B.: the same, alas, holds true for UK: the phrase has an increasingly defensive and intolerant tone to it, as in the all-too-familiar exchange, e.g. ‘Excuse me, but would you mind not smoking in here, please’—‘ ’Sa free country, innit?’ But in our modern world we may well ask not, like Pilate. ‘What is truth?’, but ‘What is freedom?’; the adage ‘Freedom for the pike means death to the minnow’ birds fair to become a c.p. of the early 1980s.
_{— Eric Partridge: "A Dictionary of Catch Phrases", Routledge: New York, 1986.}

Google suggests as early as 1848 Stray Subjects, arrested and bound over,

That chap as went in fust thar ain t nobuddy ef he has got a swaller tailed coat on My money's as good as his n and it's a free country to day This young