I've defeated Egypt twice:
1) I used nuclear weapons after I wiped out all of Israel's neighbors. Of course, there is a random chance that you cause WW3 and lose, but if you succeed, you'll win the game even if the Americans embargo you.
2) In a
Let's Play of Conflict: MEPS, I was unlucky enough to be at war with Egypt within the first two months of a game. However, the ensuring war was a stalemate that ended up hurting Egypt's economy. Eventually, the rebels I've been funding grew powerful enough for me to overthrew the Egyptian government, and therefore I won the war.
I have never beaten Egypt conventionally.