Quote:

Due process protection requires that employees have a fair procedural process before they are terminated if the termination is related to a "liberty", like the right to free speech, or a property interest.



That's right, but those are all state statutes and most, if not all, states have such laws (generally labor laws pushed through by...you guessed it...the unions) requiring due process hearings.

The First Amendment is only about government restriction of speech. That's it. Government restriction. Nobody else. You could never file a case claiming a Constitutional rights violation because there isn't one.