Any egg can hatch as a Mutant and any wild creature can spawn as one: a rare pigmentation that gives the creature an alternate colour scheme and a medal on its sheet. It is cosmetic - no stat changes - and the odds rise with the egg's grade and with your Codex.
1The medal and the recolour
A Mutant's sheet carries the medal beside its name and shows the alternate palette instead of the normal art. Everything else - moves, stats, soul bonuses - is the same creature.
2Hatching one
Every egg rolls for Mutant when it hatches - from 0.01% for a Worn egg to 0.10% for a Void egg - and filling the Codex adds a flat bonus to every roll. The egg shows nothing beforehand; you learn at the hatch.
3Catching one in the wild
Wild creatures roll on the same odds against their own grade when they spawn, so a recoloured creature on the floor is a Mutant: beat it and the egg it leaves keeps the status. Devour also carries mutant status between creatures of the same line.
Screenshots are taken from the live game by tools/tutorial_shots_b.py (dfmc-client) and regenerated after interface changes. The rules behind each screen: system-mutants.