Finchesareawesome!
Them eating chicken egg is similar to a human eating pork or beef or drinking milk; chickens are Galliformes, while finches are Passeriformes, which is pretty close to as unrelated as birds can get. Feeding eggs won't encourage cannibalism; cooked and mashed egg bear little resemblance to the eggs finches lay.
Egg is a great source of protein, fat, and minerals; it is one of the most balanced protein sources, containing ample amounts of all required amino acids. Finches are not herbivores, and unlike chickens, we do not know enough about their specific nutritional requirements to be sure that plant-based feeds are adequate. Insects are an option, but they contain a different amino acid profile and tend to contain high phosphorous but not much calcium.
My finches were semi-picky at first, but I've been able to get them to eat other stuff by mixing chopped/riced veggies with the ones I know they eat. If you have a specific dish you offer their favorite foods in, they will begin to associate anything in it with food, then you may be more successful getting them to eat new stuff. Now mine eat carrot and cauliflower, when they didn't touch it before. I'd like to get them to eat sweet potato as well, but it looks like it'll take a while.