Staggered: A staggered creature may take a single move action or standard action each round (but not both, nor can he take full-round actions). A staggered creature can still take free, swift and immediate actions. A creature with nonlethal damage exactly equal to its current hit points gains the staggered condition.
Making an Attack of Opportunity: An attack of opportunity is a single melee attack, and most characters can only make one per round. You don't have to make an attack of opportunity if you don't want to. You make your attack of opportunity at your normal attack bonus, even if you've already attacked in the round.
If they already did their action beforehand, I would think they could not. If they are staggered because of low hp's, they could use the AoO, but then would take a HP and fall. I believe it is in the rules for being at 0 hps.
...If they are staggered because of low hp's, they could use the AoO, but then would take a HP and fall. I believe it is in the rules for being at 0 hps.
They're staggered for a different reason though. I doubt the 0-hp issue would apply:
Staggered (Ex): Zombies have poor reflexes and can only perform a single move action or standard action each round. A zombie can move up to its speed and attack in the same round as a charge action.
Thinking on it... let me look up a couple more things.
It depends on what they did (or did not do) on the turn after they got Staggered.
If they acted normally then was staggered (brought to 0 hps or a target of a spell/effect), they would not be able to AoO. If they didn't do anything but a Free/swift/immediate action, then they can do an AoO, but if they are at 0 hp, after the attack is resolved they take a hp and fall.
It does say it is a single melee attack.
Am I missing something, Drakkiel?
Drakkiel is correct. However, for the followup:
AoOs do not take an action. They consume neither standard, move, or immediate action, and can be done once a round, or more, if you have Combat Reflexes. They interrupt the normal flow of combat.
You can substitute trip, sunder, or disarm for attacks, and can do so with AoOs (though whether a zombie has the mindset to do so is another matter).
Also...yeah zombies are staggered because they are zombies. If your zombie is at 0 Hp, its been destroyed because it has the misfortune to be undead.