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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 swift and immediate actions. A creature with nonlethal damage exactly equal to its current hit points gains the staggered condition.
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Take 5-Foot Step
You can move 5 feet in any round when you don't perform any other kind of movement. Taking this 5-foot step never provokes an attack of opportunity. You can't take more than one 5-foot step in a round, and you can't take a 5-foot step in the same round that you move any distance.
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Would a zombie be able to take 5 foot steps and attack?
I don't see why not. They should be able to shamble forward. It would make them just a little bit more scary.
I do not see a reason why not, it would effectively make them worthless if not.. they can charge an opponent 10 feet away but can't engage someone just 5 feet away.
There are two places that I can find in PF PRD that describe the action necessary to take a 5-foot step. Both are in the Combat section of the PRD. The first is in the text describing the picture titled Attacks of Opportunity; in this text the 5-foot step is described as a Free Action. The second is in the table titled Actions in Combat; in this table the 5-foot step is placed in the No Action section of the table. Either way if a Staggered creature can take Swift Actions it should be able to perform a Free or Non Action as well, so you have your 5-foot step, have fun.
See pages 189 and 192 of the Core rulebook.
If you don't move in a round, doesn't matter if you can only do free actions, you can take that 5' step.
However, if taking the step cost 10' (darkness, difficult terrain..), you can't move, or the effect/condition/whatever says so, you can't take a 5' step.