Places of business in the U.S. are private property as well, and generally have the right to refuse entry or service to anyone. However, the private owner of a "public place of accommodation" is governed by the Civil Rights Act of 1964 and the Americans With Disabilities Act of 1990. These set forth certain traits for individuals on which basis service may not be denied -- race, color, religion, national origin, and disability. However, members of these classes may be denied service on other grounds that are otherwise equally applied. A private business owner is entirely within his rights to refuse entry to people who do not wear masks, as long as that rule applies to everyone who seeks entrance. Previous requirements, for example to wear shirts and shoes, did not pose any legal problem. In fact, during the Halloween season some businesses disallow costume masks without incident. A business can require you to wear one, or not to wear one, as their discretion demands.
Vocal and violent protesters in the U.S. who claim a right to enter private property without complying with its proprietor's requirements are idiots. I don't know if their beliefs in non-existent rights is a universal "I can do whatever I want at any time," but they certainly seem to give that impression. And it is on par with a certain American brand of individualism that earns Americans such a disdainful reputation abroad. Here, of course, we have a sensationalist media and a childish President egging the protesters on. And there was a tragic case recently where the door guard of a thrift store was shot and killed for refusing entry to people not wearing a mask. That's a different American problem, but it explains in part why there are so many reports of this kind of exceptionalism.
While individualism is a broad American trait, most Americans will not carry it to the comical extent you see emphasized in the media. As Zakalwe notes, the extremists rarely have a coherent point. The reasons given for having a small private arsenal don't seem to extent to taking basic precautions. And the "only cowards wear masks" argument doesn't jive with their apparently fearful need to arm themselves for every occasion. (Brandishing firearms as you see these nuts do is about intimidation, not "protection.") Most Americans are courteous and responsible, but the fanatics are the ones who attract the attention and set the tone by which their more reasonable peers are judged. The objection to wearing a mask is not the inconvenience or discomfort of it, or the imagined right to patronize businesses while disregarding the rights of their owners. They object to doing something simply because they've been told to do it. As I said previously, our happy little valley is home to some notorious wintertime pollution. During the worst of it, some people commonly wear filtration masks. They do it voluntarily. But now because it's a requirement imposed by government authority or other citizens, we have mask-shaming. And that leads to a somewhat darker side of the story. Wearing masks in public is "what the Chinese do," and the ongoing conspiracy theory requires blaming China for this mess. Not wearing masks while the Chinese have to is one of the ways the fanatic fringe tries to show its defiance in the face of the designated enemy.