Could I ready an action to use Swallow Whole?
Swallow Whole is not something you can ready
If a creature with this special attack begins its turn with an opponent grappled in its mouth (see Grab), it can attempt a new combat manoeuver check (as though attempting to pin the opponent). If it succeeds, it swallows its prey, and the opponent take bites damages
You can ready a bite attack, but Swallow Whole is an ability that specifically triggers at the start of one turn. Note that it goes the same way for an eidolon or a monster, or any creature with the standard Swallow Whole special ability.