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Kirin Strike says, "While using Kirin Style against a creature you have identified using that feat, as a swift action after you have hit a creature with a melee or ranged attack, you can add twice your Intelligence modifier in damage (minimum 2)."
Since this is a swift action, I assume you can't do the extra damage from kirin strike if you hit when it's not your turn (i.e. from an AoO)? Or does this override that somehow and let you apply the damage on an AoO if you haven't used a swift that round already? I wouldn't think so, but definitely worth a check.
Swift action, so its only on your turn. If it was an immediate action, you could use it on an AoO.