I did an opera where we did something similar, and cleaned up the singer in about 1:20 offstage.
A hair and makeup team (3 people, including one wardrobe dresser) were waiting in the wings, and they used baby wipes and wet washcloths to clean up the egg (if some was in the wig, it was sometimes left and the wig was cleaned during the nightly redressing). Then they touched up the singer's makeup, which was elaborate clown makeup (we were doing "Pagliacci"). My ASM stood next to them through the process, and I announced over headset a 30-sec. re-entrance warning, and then a 5-second re-entrance warning (at which point all work was stopped and the singer took his entrance just in time). We had a towel available for the singer's hands, too. When he took his next, longer exit, he visited the makeup department for a more thorough cleanup.
The only dreadful part of the egg bit was that it got on the costume, and wouldn't come out entirely no matter what we did. Some parts of egg got onto other singers' costumes, too, and if something stuck to it and it dried, forget about it - it was horrible! The only thing I can recommend there is to get to the costumes and rinse them as soon as possible. Give a thorough check to all egg costumes - often there was a lot on the egg-er's costume, and not as much as you would expect on the egg-ee's costume. The wig suffered in this way, too, but it was our own and frankly kinda a cheap clown wig. The costumes were rentals. We also scrubbed the part of the deck immediately after each performance with soapy water and then went over it with an antibacterial spray. No problems there.
If you don't want to mess with real eggs, you can empty the eggshells (remember how you did this in grade school, when you poke a hole in each end and blow the yolk out?). It is possible, though tedious, to refill the egg with anything you want - confetti, a water/yogurt egg replacement, or whatever.
We also did not rehearse the egg until later in the process - once in the rehearsal hall (a dry run before final run through and then the final rehearsal run) and maybe every other time onstage. When you are running the egg bit in rehearsal, put a tarp down, have lots and lots of fresh towels, and provide clothes for the performer getting egged - including a t-shirt, skull cap hat, and anything else they might need to protect their regular clothes. Tell wardrobe that you'll need this t-shirt and the towels cleaned each night, but have several available through the rehearsal period. Also, be ready to have a prop towel nearby onstage for the performer who is the egg-er. They'll often need to wipe their hands on the sly after the gag.