It's not just about violence, which is a pretty rare thing even with those types. You just can't have a positive business culture if you have employees who hate other employees or some of your clients. Not wanting to have racists/sexists/homophobes at your business is a perfectly valid concern and not just motivated by wanting to "punish" people for certain opinions.
Hatreds and rivalries will always arise in businesses regardless of ideology because by definition, employees are competing for pay rises and promotions. People should not be fired for ideology unless they actually put it into practice. If the alt righter starts refusing to work with and verbally abusing black employees, then there is, of course, good reason to fire. But, some stupid memes they posted on facebook is not good reason to fire. Many people act very different online than in person anyway.
Do you seriously think some hardcore stormfront-esq racist would provide equally good customer service to a black customer? Of course not, thus they're an inferior employee to someone who would.
Yes, they would because if they didn't, they'd get fired.
Once again, going back to the original point about sympathy, even if it was entirely ok to fire people because they are alt right, that doesn't mean you can't be sympathetic to someone who is going to suffer for the rest of his life because of his dumb opinions, that he may not even have anymore, people's opinions change.