I don't know if they should be able to but in the USA at least can't you fire anyone for any reason as long as they aren't part of a protected class? I don't think there are any laws against political discrimination, are there?

Political affiliation or beliefs are not a protected class in the US, so yes you can very much get fired for that without recourse.

Supporting Trump, no.

Then again, if you were filmed taking part in a violent insurrection at the capitol building, please don't bother coming back to work if you work for me.