I say a dog. Cat's like to curl up on your lap, which i can't really see with Jack. He probably has a small dog, one that doesn't require a lot of attention.
Jack is a cat man. He needs someone as independent as he is. Look at the women he chooses. They're definitely more catlike than doglike. I think this preference extends to pets.
I think he has a cat actually. In the episode in season 4, when Sydney and Vaughn visit him in his apartment you can se se a cat bowl or something with [I think] cat food in it.
Yes, the prevailing theory seems to be that Jack has a cat, and the visit to his apartment in "Mirage" started it. Some say it was probably just a stray he started feeding. But I think liking cats says a lot about a person!