crazychic, In my opinion there could be several reasons why you have a egg eater.
There are a number of cures for the problem and culling the unfortunate hen with egg on it's face is not a solution in my opinion, because any hen will eat a broken egg.
1. collect eggs often, broken eggs because hens to fight for the same nest box is common.
2. Darken the nests to provide privacy for the hen to lay, by hanging a material in front of the entrance with a slit for the hens to enter, which they will.
3. Increase the amount of protein in the diet. Treats like dry cat food, 30% protein,
black oil sunflower seeds 30% protein are a good start. Eggs have alot of protein and hens have the ability to know what their diet is lacking.
4. To assume a hen with egg on her face makes her the egg eater can result in the same problem to continue, in my experienced opinion.
bigzio