In the singular usage it more generally refers to a person holding French citizenship. More specifically, to a Frenchman holding an opinion that is deemed (by the listener) as unsupportive - especially regarding foreign policy or military matters.
It refers to the supposed cultural habits of Frenchmen - cheese eating and surrendering when any slight agressive action is taken towards them or their country. It is considered perjorative in any sense.
Scene girl #1: Like, RaWwRr! Let's go to the br00tal show.
Scene boy that looks like a scene girl: I'm going to show off my face-eaters and look fierce.
Indulging in an exorbitant quantity of food, outside of any meal time, and hiding the fact that you have just scoffed an amount of food that would feed a small town in Kenya.
It's 4pm and Louanna has once again found herself in the stationary cupboard in the office, hiding away from her colleagues, as she stuffs her face with a large serving of McDonald's fries. Louanna is clearly displaying the behaviours of a shame eater.