Do thrown eggs deal damage?
Solution 1:
From the Minecraft Wiki article on eggs:
Players are able to throw eggs at mobs and give the same effect as if the mob were being damaged by a weapon. They deal half a heart to Ender Dragons (except when hit in the head), but no damage to other mobs. Eggs can also be fired from dispensers and are affected by gravity. Throwing eggs at neutral mobs will provoke them.