I think as chinadoll mentioned you will need to add oils to up your fats..I don't think just eggs will do it.
Egg salad perhaps with mayo, eggs fried in butter etc... since as you mentioned protein is a poor energy source, and since you are exercising it's important you keep your fats up.
It doesn't take much oil to keep the fat up....you may want to open an account at
www.fitday.com or
thedailyplate.com these are online free food diary tools that will let you know your %'s of protein/fats/carbs you are having for the day. They're very handy tools.
Reading the Atkins book is also a very important part of this woe. That way you will know the rules of induction, and OWL when adding back the additional food groups.