The only thing that an individual should be judged on at work is actions/comments/etc. made at work.
If you'd like to argue otherwise I'd hope you at least have something more formally define-able than 'he was trending towards being a neo-nazi'. I'd also expect you to be able to point to an authority for defining these ideological limits other than the currently loosely defined group-think.
Please answer these few scenarios:
-Should someone believing and actively promoting pro-life stances(outside of work) be fired?
-Should a casual narcotics user on weekends be fired once it's found out he/she uses?
-Should someone be fired for having a confederate flag hanging in their living room? What if they're black?
They also scream white power around the campfire when drunk though so perhaps not typical.
I can perfectly imagine any number of anti-democratic, politically obsessive people who would think that way, yes. They are certainly welcome to hold that view, but if the rest of us want to maintain a cordial and pluralistic society instead of one of constant political warfare and discrimination, we need to oppose those people at every turn.
You should care if your coworker is neo-nazi or if he is supporting Islamic State or Westboro Baptist Church.
The only diagreement is whether Eich was closer to the former or the latter. I can perfectly imagine people whose rights he wanted to take away to think he's the latter.