With my first team, my 3 pets became rare at level 13 or 15, I do not remember, by themself. I have not gave them any stone.
Did you had the same experience with some green pet that you have up?
My first team waas:
Crimson Whelpling P/S
Disgusting Oozeling H/B
Magical Crawdad H/H
They are still displayed as "white" (but they are green) in the armory, but all are blue in my codex.
Is it automatic past a level?
maybe it depends on the number of fight, or number of fights you win?
Please, let me know.