Haste has the following line:
When making a full attack action, a hasted creature may make one extra attack with one natural or manufactured weapon.
The staggered condition states:
A staggered creature may take a single move action or standard action each round (but not both, nor can he take full-round actions).
In the rules for Full Attack we see:
If you get more than one attack per round because your base attack bonus is high enough (see Base Attack Bonus in Classes), because you fight with two weapons or a double weapon, or for some special reason, you must use a full-round action to get your additional attacks.
Since Haste only grants it's extra attacks on a full attack which is a full round action, and the staggered condition prevents full round actions, they do not get to make the extra haste attack.