An egg's yolk and shell can do serious damage to automotive paint. The broken shell can scratch through the clear-coat layer or deeper through to the base-coat. The acidic yolk and whites can eat through multiple layers of paint and stain what they don't destroy.
In order to minimize damage, the egg must be removed as soon as possible. Once the yolk dries, removal becomes difficult and care must be taken not to scratch the paint further. Spraying the egg with white vinegar is one method to help in safe removal. Automotive clay and detailing wax can help fix damage that is not serious enough to warrant a professional touch-up.