If I were an employer, I would invariably also have employees who are not "white." Such public disclosure by a white supremacist would create an unhealthy environment not conducive to productivity and profitability. There would be discord. People can have their own views, but if they are repugnant views then they should expect to be held to account.
Wholeheartedly agree! However, the decision to go after the employee should be made by the employer. It shouldn't be an angry mob that chases down someone's real address and the like because of a comment. In the case of white supremacy, that's easy. But what if it was a person standing up for a decision Trump made? A bunch of folks who hate Trump then find this person's view despicable and pressure the company he works for, or threaten the family, etc. It's not always an easy thing like white supremacy. Also, the other side of the coin doesn't see a whole lot of action. Show me one African American who was fired for saying "kill whitey!"
If you are too scared, worrying about drawing lines then you will never draw them and then anything goes. That's not how civilized society works. And if you can't draw a line at, say, white supremacists, then what are you good for? You can have your views but be prepared to take responsibility for them and hope that there are enough like-minded people who will give you cover. Because not everyone will.
The Constitution already drew the line. Hate speech with violent intention is not allowed. Everything else goes. Our country (the US) was founded on it.