No, they don't paint, but they help herd the eggs to the flowers that paint 'em.

[He concentrates, laying a paw on the ground, a small blue flower sprouting up from the snowy ground. He bats a googie towards it, and the flower releases a puff of blue at it, covering the egg in a mottled paint]

There's more to it than that, a river that gives them swirls, vines that give them patterns. It's a big operation.