Devil’s advocate: The First Amendment only says that the government can’t limit your right to “free speech”, a right which the courts have interpreted broadly. Nothing in there regarding what a company can do or not do.
And, as Charles Evan Hughes so aptly put it, the First Amendment doesn’t give you the right to shout “Fire” in a crowded theater.
Can your company, in legal terms, limit your rights to free speech? Tough to answer. I live in California, where employment is “at will”, meaning the company can fire me for any reason. We have both state and federal statutes prohibiting discrimination, and I would have to prove that the company was discriminating against me when it fired me.
Simply “speaking my mind” is not covered under any discrimination laws I know of. “Whistle-blowing” is, but I don’t think the OP is talking about that.
It’s both unfair and stupid for your company to place these limits on your outside speech or force you to hype the company (or both?). Unfair, in that the company should focus on individual behavior. Stupid, in that it just makes the company look backward and oppressive, both to its employees and to the outside world.
Still, it’s “legal” in the sense that any action they take against you would be legal (subject to the limitations I’ve already mentioned), including firing you.
If you, or anyone, really wants to be anonymous, you can find better ways of doing it than e-mail or Facebook! Alas, most of them involve printing things on paper, which doesn’t get you nearly as much distribution.
If you think about it, the Internetz is not at all anonymous. The best way to make your ideas widely known is to post them on a frequently-visited website. But unlike newspapers, which in olden times printed opinions and then fought to keep the author secret, websites lean in the opposite direction: “Oh we’ll get sued, so we keep track of who you are and had over the information if someone shows up on the doorstep with a warrant.”
And you can’t hide. Your company is probably searching for you even now, trying to catch you doing wrong. As I said, stupid but perfectly legal. You can do two things: go work for a good company, or get everyone at your company to violate the regulations.