I have done this in the past on siding and it worked very well, it's worth a try:
Mix up a batch of warm water and white vinegar, the mix is 50/50 [On brick or stone, unpainted, I use the vinegar full strength].
Take an old rag and soak it in the water/vinegar solution. Lay it over the egg splatter and hold it in place with a 1x2 or pole for about 15 to 20 minutes.
This will get the bulk of it off.

Next you need some laundry detergent that contains a protein enzyme. Make sure it has this enzyme or it won't work.
Mix the detergent with warm water to form a sticky paste, apply this paste to the remaining egg stain and all those other little splatter spots around and below it. Often their is a long streak running down the wall from the egg.
As soon as the surface of the paste starts to dry, wash it off with a scrub brush and warm water.

Hi, Thanks so much for your solution. I will tell you that I did try the white vinegar idea yesterday, but did not let the vinegar soak on the stains for 20 minutes. Basically, I applied it to a towel and tried to quickly rub off the stain which didn't work, so now I'll try and let it sit as you suggested. Will your idea work on a roof? You said you used it for siding..
Since this is a roof, I may just use the vinegar at full strength. Vinegar will not stain or remove the color of the shingles will it? Also, as far as the detergent.. Can you recommend one that has the protein enzyme? I currently use Tide. Does that have it? Thanks so much for your help.

I can't speculate whether vinegar will stain your roof or not, might should test a small spot on the back of the house first.

I actually used Tide and made a paste by adding liquid "Era" detergent which has enzymes that will break the egg protein.