I am an employer myself so yes I am sure about this. However that doesn't stop companies from find other reason to let the person in question go. In my professional experience very few employers look into a person's personal life via Facebook or Twitter or whatever. Usually because your average Joe on Twitter is going to have some random ass Twitter handle like "Bigdick666" And you would have no way to link that to James in Accounting unless he logged into that account with work property.You sure about that? That companies can't and don't put in their rules for employment "No hate speech, racism, etc"? Because support for Nazism fits into both. Not to mention quite a few of your coworkers might/will decide they don't want to work with you, leading your boss to tell you "You're not a good company fit. Bye". You act as if companies need to justify their firing other then "We don't need want you here. You don't meet company expectations". You also imply the US government really gives a shit about rights of workers over corporations and businesses.
There's also the everpresent "Bad PR" angle. Because the moment X gets outed writing "Hitler should have finished the job" on a message board, which starts tending on twitter, which makes the news, then gets picked up by somebody working for <Company X works for>, it's a short step to "You don't represent the company values we believe in. Pack up your shit". Because otherwise you get the "Man, <Company> knows X is a Nazi and keeps them on. Fuck <Company>!" PR backlash.
As a result the only time this would even come up would be if it spilt into the workplace environment. At which point the person is likely to get a warning and a mark in their file with no further punishment unless there was complaints from other co-workers of constant disturbance.
But usually racists, or nazi's aren't walking up and down the cubicals at work shouting "kill all Ompaloompa's" Or whatever.
I mean when you consider the sheer fuckload of people on social media, think of how rare it actually is for you to hear when someone gets caught being "hateful" online. Usually you have to do something absolutely bonkers to get caught and suffer from the reprocussions of such behavior.