I also don't think there would be a "brown scare" equivalent to the red one as such- remember, there was paranoid anticommunism in the US before anybody had even heard of fascism. Red scares tapped into deeper, more established trends in US political life.
I think what would happen in a neutral US-Nazi Europe timeline is a generalized left-wing populist, antifascist attitude. I think that the sort of ambivalently pro-Fascist language that was popular among the right-wing and rich in the 1930's would start to be shouted down in the media more and more aggressively, until there was an effective, informal tacit ban on floating certain ideas or advocating certain pro-fascist policies.
So I don't think HUAC would exist, or that it would be dragging lucky lindy in front of it (although the FBI would certainly be watching) but anybody speaking out in favor of the german-backed military rebellion against the Tudeh government of Iran in the 50's would be shouted down by feistier members of congress and the New York Times and generally be seen as an un-American asshole.