I've always had strict rules about employees conducting themselves in a professional manner in the public eye. In this case, her offense is apparently even more egregious because she lists the name of her employer on her social media accounts. Posting a photo of yourself making an obscene gesture is a no-no for any responsible employer.
Listing the name of her employer in a Facebook posting that includes this does cross the line. She cannot claim to be speaking for her employer, nor risk the resulting damage to his business.