Should businesses, big or small — from Walmart down to a family-owned grocery store — be legally allowed to kick you out because of what you wear or say? For example, if someone wears a Nazi shirt, a KKK shirt, or has a Nazi tattoo on their body, should a business be allowed to kick them out? And, just to be fair, this applies to both the left and the right. If someone walks in wearing a pro-Palestine shirt or a Black Lives Matter shirt, should a business be allowed to kick them out too?
Up to the point of protected speech, yes.
Free speech doesn’t mean you can say anything, anywhere, anywhen. It means that you can expect to be free from persecution or prosecution based on expressing a range of beliefs via speech/press.
There should be a difference between a truly private property, as in one’s home, and a business property, but that’s a different issue. Even with businesses being “private” property, they still have to toe the line with protected speech, and most of the examples you gave would fall under that. The problem comes in litigating the violation