Find a stationary store that makes rubber stamps. They can make one from your design or a design that you find in print so long as it's a strong black image on clean white. But there's a good chance it will cost somewhere in the vicinity of $17. (I paid $30 for my carton stamps but I LOVE them.)
Tell them you don't want it mounted. ...or want the mounting materials but don't want it assembled. Then you'll have to locate some backing for the rubber die that's far softer and more malleable than the typical rubber backing they'll supply so that the die can form or mold around a curved surface.
Once you get all your supplies, use rubber cement to affix the rubber die (with the image) to the rubber cushion/backing. And use more rubber cement to affix that to the plastic or wooden mount.
If you want a decent image you're going to have to limit yourself to a small image or a linear one that's not at all tall and develop a skill at "rolling" it across the curved shell without blurring your image.
I bought a tiny one that wasn't bigger than the tip of my pinkie and had a hard time getting decent printed images. I ended up getting larger and more interesting stamps I could print on the cartons.
And BTW, if you're going to print on eggs and distribute the eggs to anyone else there are restrictions on what inks that are non-toxic. ...tho I'm not personally convinced that a tiny bit of ink on an unwashed sealed egg would be an actual problem.
