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.
The way America works is if you can be fired by being a Nazi, you can also be fired for being a member of BLM.
You don’t have to be fired specifically for being a Nazi (or BLM). You can be fired or other reasons, like creating an uncomfortable working environment for other employees or clients/customers, or for a personality conflict. And that’s to start with. It might be shaky, depending on the individual circumstances, but an employer isn’t absolutely obligated to employ someone if they are disruptive in some way.