So you want to use a robot to draw on eggs and other spherical objects? The Sphere-o-bot can do the job. This is a 2-axis drawing machine that can be used to decorate balls and eggs. You are going to need an Arduino Leonardo board, JJRobots brain shield, stepper motor drivers, 1x SG90 servo, and a few other parts to complete this project.
Access to a 3D printer is ideal. You can program the board using the Arduino IDE. More info is available here.