A lot depends on how long the egg had been on the car, and how old the paint is. If you got it washed off right away and it left a stain you might be able to get it off with a fine grit rubbing compound. If that will not do it you may need to use a high speed buffer with a course compound, then followed up by a real fine compound and polish. If none of this takes care of the problem, IT'S PAINT JOB TIME.
I have only had to do one egg repair. I got the car the day after the egg hit it. I found that the shells are very sharp, and actually cut the paint. I would recommend close inspection with a magnifying glass if you don't see paint breakage. If you do see that the paint has been pierced, it needs to be repainted in that area.
Get the moisture back into the egg. Wet the area down, take a clean, wet rag and lay it over it, keep it wet for a half hour and check it. keep it wet until it wipes off.
Hi i had your problem to what i did was buy some "Bugs B gone" its in a white and red spray bottle you get it from auto zone i think it was ten dollars but most autozones sell it for six dollars its really for bugs but will get all the eggs off your car and will not take any of your paint