Male students in pre prom photo pictured giving the Nazi salute

I think you hit the mail on the head. There can be no “formal” punishment from the school or government body as it is protected speech. If society enacts legal consequences (public shame, loss of reputation, etc.) that is the punishment but any formal punishment would seem to violate the 1st amendment. Keep in mind, as abhorrent as the picture/action may be, no crime was committed here.