Normal: "A fiend that once roamed the Calm Lands and subsisted on chocobos. After an earthquake trapped it underground, it developed into a far less picky eater." Oversoul: "A fiend trapped beneath the Calm Lands. Oversouling must have made it grumpy, because it now selectively targets certain dresspheres." — Scan
Equipping a Star Bracer prevents all of Anything Eater's spells from hitting the party members. The only thing to worry about at this point is its physical attack, which can be taken care of by using a Light Curtain or using a dressphere with high Defense, such as Dark Knight, or using Berserker for their Evade & Counter ability.
If Anything Eater is Oversouled it is best to avoid having all three party members with the same dressphere on at the same time, and make sure to switch dresspheres whenever the message "Me no like dressphere!" appears. This ensures the boss cannot attack anyone that would be of that dressphere if they all are different.