Right-wing supremacists ought to be driven out of the military.
Yes you are right. So should any other person who behaves in a disgraceful, unacceptable and socially intolerant manner.
A nazi or a Nazi supporter, is obviously vile and unacceptable.
A white supremacist is also vile and unacceptable.
A KKK/any other hate group is also vile and unacceptable.
So is hamas supporter, taliban supporter, any anti-semite, anti-white (yup its a thing), christian hate/anti-christian are other examples of hate groups that are equally vile and unacceptable.
Curious, what you describe as a "Right wing supremacist"? Right wing to me, is people who lean politically to the right. So would left wing supremacist be ok? I am not being contrarian but as someone who investigated troops who turned out to be committing acts of hate (had to turn the files over to the MPs), hate comes in all forms, not just "the right". Careful of your dialogue. Its slips easily from right wing is unacceptable to conservatives are unacceptable.
My own personal experiences I have encountered these types of hate during my service in the CAF (1990-2018)
-Many anti-black (from dirty humour to outright hatred)
-Many anti-semite (Jewish bashing I found problematic with many greater Toronto area reservist of a certain religious group, take a guess which one and it really disturbed me)
-I experienced it myself a few times, usually in dark humour (I have a German surname so I had more than my fair share of Nazi comments/third reich, etc, ironic, my best friend is Jewish and his grandfather survived the concentration camps in Poland)
-Anti-Muslim (this one picked up after 9/11 but was the easiest to reign in)
-Anti-Gay (More so earlier 90s and 00s, by 2005 or so, almost everyone seem to no longer give a shit)
-ANti-Italian (not that bad but some bone heads made comments, my wife is 100% italian descent)
-Anti-Sikh (Especially early 90s, this was actually some of the worst comments I heard)
-Anti-native (I met a few guys here and there who had nothing but disgust and bigotry for it)
-Anti-women in combat arms (again especially in the early 90s)
When I was a troop, other than not participating or walking away, not much I could do. By the time I became a Master Corporal, I had decided I would not tolerate this behaviour period. I surprised a few people who thought they could come up and "make some hateful comments, thinking I was of that type, big surprise, I would shut them down or report them CoC.
I was also surprised in 2000, when a black NCO I knew was charged for calling one of his black peers the N-word, he thought he had a free license to say it in jest.