You might want to check Veggie Grill in Irvine near UCI. The food is great and I remember that they have a dessert section that includes cake. You might want to call first to make sure its eggless. http://www.veggiegrill.com/menu.html
This restaurant is entirely vegan so yes everything is meat-free, dairy-free and egg-free. I really like their carrot cake, which is the one they usually have. I heard they had chocolate cake too, but I've never seen it.