You don't say what kind of shingles you have. If common asphaltic type, I suppose that a pressure washer on low setting or a scrub brush with soapy water.
Eggs are protein and protein stains can be released with enzymatic cleaners like in some laundry presoaks. The science teacher might be able to give you some hints.
Wood shingles might be more difficult because they are porus and can't handle pressure washing.