“they would fire her for misusing their brand identity.”
Definitely. I think that has held up in court before. Seriously jeopardizes their ability to conduct business.
Of course it has worked the other way, too. People posting on their personal sites and being fired for that.
It was her personal page, from what I understand, but she may have said something like “as a Delta employee,” or something that made it look like Delta was authorizing whatever she said. Which apparently was a left-wing lunatic rant.