I'm late to this party, but I think that he should not have been fired. If his Nazi sympathies aren't affecting his job performance or his relations with other employees, then he deserves to stay on the job.
He wasn't fired because of job performance. He wasn't really even fired because of his employer's views of his beliefs. He was fired (as were his spouse and brother-in-law) because the owner of the restaurant received threats of violence and death if they weren't fired. His personal beliefs weren't affecting his job performance, given that until the article was published, his employer did not know he was a white nationalist.
That puts this into a pretty tricky spot for me. On the one hand, I strongly uphold each person's right to both belief and speech, regardless of how odious I find those beliefs to be. In this respect, I am opposed to him losing his job as a result of his beliefs, let alone the secondary effect on his family. On the other hand, I can definitely sympathize with the restaurant owner for his actions, as I would not expect the owner to suffer violence or threats in order to protect the rights of another person.
At the end of the day, to me, the blame lies with the mob mentality that it's acceptable to threaten others in order to squash their beliefs. That view is profoundly undemocratic.