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
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 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.