One could be fired at will, in parts of the country, but if it falls under the protection of one or more of the Amendments, one better hope the former employee can't be bothered to bring suit, due to lacking time and money.
That's just wrong and silly.
The amendments refer to what government can do to citizens, not private business in an at will state. You cannot sue a business over the First Amendment or the Fourteenth, they apply only to what government can do. Try suing over an employer over not letting you have a gun at work under the Second Amendment.
I fired a lot of folks in 40+ years in business in everything from layoffs to 'with cause', every time working with our General Counsel and never got sued although I did get threatened fairly often.
If I wanted to can a Nazi/White, Supremacist, General Asshole (fired a few of those) it is easy provided no union or civil service protection.
Eliminate the position - easiest of all. No cause needed, just don't need that position at that spot at this time.
Document a violation of the Company Guidelines/Ethics Statement - (easy Nazi killer)
Document any and all hostile/racist/sexist/sketchy statements or behaviors - (hard to believe a Nazi could stay in bounds)
Set goals seemingly reachable but actually impossible and then document failure. Takes longer but kill them in a quarterly or mid-year review then put them on a performance improvement plan they cannot meet and then can them all nice, neat and documented.
Bottom line. If your employer wants you gone, you are gone and unless they are an idiot, you aren't winning anything in court.