'For breakfast' indicates that you're referring to the main ingredient of the breakfast. If you say 'at breakfast' it's just one of the components. So 'eggs for breakfast' suggests she doesn't eat much else at that meal, whereas 'eggs at breakfast' either indicates that she eats eggs at breakfast but not at another meal, or that eggs are just a part of her breakfast.
Oh, I see. In that case, if I wanted to say that eggs are just a secondary part of the breakfast, I would maybe say "Jem jajka do śniadania".
But I can't think of a better way to say that I only eat eggs at breakfast time instead of other parts of the day than to simply say "Jem jajka tylko na śniadanie" (or, again, "do śniadania", if the eggs are not the most important there).
Anyway, all of that is if somebody actually cares about the main-secondary distinction. You might hear "na śniadanie" for either situation anyway.