Sorry, I don't know the answer but I cannot imagine that you could leave the eggs out without replacing them with some kind of egg substitute as a binder. It is my understanding that the eggs are an important part of your cake's structure! Hopefully someone else will see this and chime in with more details. I also normally see oil substituted with SOMETHING - even applesauce in some recipes. Perhaps it is indeed the yogurt in this case but I can't be sure 
- sorry I'm not superhelpful but I just wanted to chime in my reservations about leaving out the eggs!
I sub oil w/ applesauce. Are you making it for yourself? If so you can get a cake mix & use 1 can of diet soda. With eggs, use whites (1 extra) instead of the whole eggs
Along with the substitution of applesauce for the oil, you can always use egg sustitutes like Eggbeaters if your goal is lowering the fat.
Also, you can use pureed prunes in chocolate cake instead of the applesauce. You will never know they are there. Baby food prunes work great.
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