To have "egg on your face" means to be embarrassed. Like "I
messed up completely and now I have egg on my face." If you "had
egg on your face" it means that you were caught doing something you
should not have been doing. The image is of a fox or cat who has
been chasing the chickens and eating their eggs, trying to pretend
they are innocent when they have the egg all over their face where
they were eating.