To be extremely embarrassed. Usually the embarassment is the result of one's own actions. Not sure of the origin, but it
may come from when clowns are at a circus and they have
eggs thrown at them because of their goofy acts or when actors had
eggs thrown at them performing in plays a couple of centuries ago.
"I am sure you still have egg on your face after causing a scene in the
mall and later finding out that your boyfriend was in the
mall with his neice from
Chicago instead of another girl".