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".