Chalkboard Paint

You can make these cute eggs that have been painted with chalkboard paint.

Nowadays, you can buy small cans of chalkboard paint for a decent price in the craft or paint sections of many, many stores.

Basically, after you boil the eggs (or blow out the innards) you paint them.  Once dry, you can draw on them with chalk.  Prop up the eggs with papertubes whilst you are painting.

Alternative:  Black acrylic paint, and paint markers (permanent designs).  Especially if you are blowing the innards out of your eggs and want to save them for other Easters.