How long has it been on the paint, and has it been baking in the sun? From what I understand, the enzymes in the egg will damage the paint pretty quickly.
However, last Halloween there was a little egg I found the next morning on my wife's van. I washed it off promptly (when I saw it) and it left no etching.
If it has been on there for days (especially baking in the sun) then you're prolly gonna have to polish.
Wash the car ASAP any soap will do at this point, then inspect the area to see if you have any etching, I would then clay those areas, and use a paint cleaner, chemical based polish is what that is...if the paint is etched you might need some wet sanding..if not then the polishing will do the trick, then apply some good paste wax
Once you get it washed off inspect the area very carefully. One nasty byproduct of an egging is the broken shell actually cutting (micro-chips) into the clear coat. The damage sometimes looks like a large fingerprint.