Distinctly untrue. The American Declaration of Independence denounces the Quebec Act. American efforts in Quebec were confused and half-hearted. If Quebec had been conquered by the US, it is likely the Anglo oligarchy of the US would’ve revived the restrictions on Catholicism and Quebecois culture that the Quebec Act abolished.
This is actually probably the major one thing in Quebec history that I'll differ from Willdavie on.
Quebec would not have fared better under American control than British, and I'm highly skeptical that it would've been actually independent. Being one province of Canada is easier to put your foot down than being an American state - I have no problems with the American invasion having failed.
The American invasion didn’t deserve to succeed, either. It was a classic case of overextension when they really should’ve focused on defending American lands. Neither Montgomery nor Arnold were equipped to carry out a siege and coordinate the logistics it would’ve taken to take Quebec from the British. If America couldn’t invade Quebec, there’s no way they could’ve governed it.
I doubt that america would have clamped down on Catholic and quebecois culture in the ways you suggest as America had already adapted significant cultural cleavages into its governing structure by 1776 (like Baltimore) and a French controlled area with the purchase of Louisiana under Jefferson. What is more likely is that American government would’ve been ill-equipped to manage quebec as it was significantly larger and more populated than other territories under their control. American mismanagement would’ve likely doomed the Quebecois to a miserable economy and raised cultural tensions that could’ve resulted in their independence by the 1812.
Again, the prospects of America even succesfully invading quebec (or annexing it) are far-fetched. Even if America had conquered Quebec, they would’ve had to return it in negotiations to the British or hand it over to the French, who would’ve mismanaged it similarly to the Americans and condemned the province to political and economic collapse during the French revolution.