The real issue here is that it's not actually about the AI part - it's about whether your employer decided the fallout was damaging enough to justify letting you go. In at-will states (which is most of the US), they don't need much of a reason at all, and public backlash against a CEO is pretty much the perfect storm for that. If customers, clients, or the internet in general started calling for her head, the company's liability calculus shifts fast. They're protecting their brand, not making a judgment call about fairness.
One thing people gloss over though: what matters most isn't what you posted, but how *visible* you are. A regular employee posting something dumb on a private account? Way harder to tie to the company's reputation. A CEO whose face is the company's face? That's different. The moment her personal image became tied to the organization's credibility in people's minds, she became a liability. Companies will absolutely use that as cover to fire someone they wanted out anyway, or genuinely worry that keeping her costs them business.
The deeper pitfall is assuming there's some rule book that protects you here. There isn't one - not really. Your employment contract and your company's social media policy (if it exists) are what actually matter, not some universal standard about what's "fair." So before you post anything that could blow up, it's worth asking whether your employer could realistically connect it to themselves. If the answer is yes, and it's anything controversial or deceptive, you're rolling the dice.