I think it must. You can make a case that UCMJ and waiving of rights are limited exceptions that must be allowed under the first amendment for the government to do its work, but as these rights are fundamental, I think it is the government that must justify restrictions on any citizen's right to free speech. And it is our system which requires that those restrictions be as limited as possible.
IMO, the First Amendment is in play because any restrictions on speech must be justified and limited.