Freedom of speech is an unalienable right, but if I owned a business and a bunch of lefties came in and started telling all of my customers how evil conservatives are, I would most certainly have the right to throw 'em out.
Of course you do. Not only that, but you would have the Right to expect the law to arrest them if they refuse to leave. You would also have the Right to recover any damages the tresspassing group inflicts. Trespass laws have been a foundation of Western law for hundreds of years.
But if you have a business that is open to the public, you can't simply post a sign saying "no (fill-in-the-blank) allowed" AND expect the law to enforce it. The exceptions to this (such as "no soliticing" signs) are few and far between. And for good reason.
It should definitely not be a criminal act for a legally armed American to enter a public business. But a lot of so-called "2nd amendment supporters" seem to support that.