As far as I know, this guy hasn’t killed anyone or committed any acts of violence. Until he breaks some law, he has as much right to be a nazi someone does to be a member of BLM. That is the way America works.
You can absolutely be fired for your political beliefs in America – that’s how America has worked every day since its inception. Whether that’s a good or bad thing, and whether we should change the way America works so that you can’t do this, is open for discussion.
Again, being a Nazi isn’t equivalent to being a member of BLM, in the same sense that being a cannibal is not equivalent to being a vegetarian.