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?
it’s a free country
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.