Man, there's a lot to unpack here.
Yes, I support worker's rights. I also support employer's rights to not have to employ someone who's making their company/organization look bad.
We're not talking about actions in private. Posting stuff to Facebook or Twitter is public.
This doesn't apply just to professionals. If you're working the fryer at McDonalds and you start posting "man, some of our customers are STUPID" type posts to Twitter, if they can tie your Twitter to your identity, that'll get you fired, even if you're "off the clock".
"If the employer were the government" is a ridiculous "if". They're entirely different things, so saying "if context was totally different, the way things went would also be different" is nonsensical. Especially since badmouthing people on Twitter would still get you fired from government jobs, because First Amendment stuff doesn't apply to that.
There's no demonstration that this was a "false accusation", and regardless, responding to an accusation would never justify acting out in a bad way like this.