I'm just curious on if you can really be fired for a Facebook post that makes no mention of your employer? Yes, you can. Because there is no law that says you can't be.
Are they allowed to terminate on their own assumptions? Yes, they are. Because there is no law that says you can't be.
Where is the grey area? Oddly enough, depending on EXACTLY what you post, you MIGHT have more protection if your Facebook post DID make mention of your employer. But that is very, very, VERY fact specific and you absolutely should not take away from this that you can't be fired for a Facebook post that does mention your employer. Because you can, mostly. But there are a very, very, very few situations in which you can't.