NAP has a creature equipped with Captain's Claws.
AP casts Press into Service targeting that creature, and attacks with it.

What happens?

My assumption is that NAP (still being the controller of the equipment) gets the 1/1 token, which is tapped but not attacking.

Hello,

Captain's Claws does not give the ability to the creature. It's a trigger from the equipment itself. If Player A controls the Claws attached to their creature, and Player B gains control of the creature and attacks with the creature, the token will enter the battlefield under Player A's control (since Player A controls the Claws). The token will be tapped, but will not be attacking.

506.3b If an effect would put a creature onto the battlefield attacking under the control of any player except an attacking player, that creature does enter the battlefield, but it’s never considered to be an attacking creature.

Nathan Long
Magic Rules NetRep