I would try polishing compound, I am sure it will never completely be gone.
found this also:
STEP 1: Get to the egg mess as soon as you discover it. Do not let egg sit on your car, even overnight. STEP 2: Remove by hand the egg shell pieces that are not dried onto the car. STEP 3: Wipe up as much of the egg as you can with a soft towel, being careful not to scratch the paint with eggshell remnants. STEP 4: Soak a towel in a solution of half warm water and half white vinegar. STEP 5: Place the saturated towel on the egg stain, letting it sit for 15-20 minutes. STEP 6: Remove the wet towel, wipe away remnants and dry the area with a soft, dry towel. STEP 7: Take your car to a body shop if the paint or clear coat is cracked, crinkled, scratched or bubbling up. You may need to have to have the car spot-painted or completely repainted.