I was egged last night and just found something that works to remove the hard, dried egg yolks & whites: CLOROX STAIN-OUT (laundry stain remover). This product does NOT contain bleach - rather it is a protein stain remover.
I applied the product (saturated) to my garage door (painted aluminum) and cement driveway. In a matter of minutes I was able to wipe the garage clean with a wet washcloth. I used a scrubber brush to scrub the egg out of the cement. (I had first wet the surfaces with my garden hose - then applied the product.)
Perhaps it would work on the brick with a scrubber brush ...? I would think it would also work well on your windows/screens. Surely, check on a small area first to make sure the product doesn't do further damage to the egged surfaces.