So the question becomes, as it so often does after these firings, were their employers legally in the right for firing them for merely being a neo-Nazi or attending a white supremacist rally?
And the answer is, as it so often is to any legal question, it depends.
The General Rule
Most employment in the United States is at-will; meaning that employers don't need a good reason to fire an employee, they just can't do it for an illegal reason. So, free speech and First Amendment arguments don't work against private employers -- those are protections against government censorship.
That said, even private employers can't discriminate based on race, religion, sex, national origin, age, disability, or certain union-related activities. But note that federal anti-discrimination laws do not include protections for political affiliation.
As legal commentator Eugene Volokh points out, however, many states, counties, cities have statutes barring discrimination based on off-the-job political activity, like attending a white supremacist rally. These statues can vary wildly in terms of the kind of behavior to which they apply, from "political activity," speech, affiliation to election-related activities and even any legal off-duty behavior.
Many of these statutes, however, afford employers some leeway if protected off-duty activities affect on-the-clock business, either via intra-office morale or the company's bottom line. Another legal question exists over whether an employer should be forced to fire an employee over bigoted beliefs or speech on the grounds of avoiding harassment or a hostile work environment, and that question becomes even thornier if it involves off-the-job speech.
Want to fire a fascist? Check with an experienced employment attorney first.