Chickasaw, Alabama
Looks like there is a precedent for the Bill of Rights being upheld even in
privately owned towns.
As a general rule, the owner of private property is free to restrict expressive activitites of others on the property. You are under no First Amendment obligation to admit people into your living room and then listen to them blow off about any topic of their choice. Similarly, an owner of a restaurant has no duty to allow persons who dislike the food she serves into the restaurant so the person can annoy customers or discourage others from eating there.
Yet, almost every rule as its exceptions, and this rule is no exception to that rule. In 1946, the Court considered the issue of the First Amendment's applicability in Chickasaw, Alabama--a company town owned lock, stock, and barrel by Gulf Shipbuilding. A Jehovah's Witness came to Chicasaw and began distributing religious literature on a street corner. She was told to stop her activity. She refused, and was tried and convicted of trespass. The Court reversed her conviction, concluding that Chicasaw was the functional equivalent of a municipality, the residents of Chicasaw citizens of Alabama, and that the First Amendment fully applied to expressive activities on the company-owned sidewalks and streets of the town.