They are actually fox hunting dogs. Although, dogs that are bred to hunt one type of game, will usually instinctively hunt many different types of animals. Since Jack Russell's are supposed to chase foxes that have gone into their dens out to the open, they will be more likely to want to hunt other animals that are inclined to go underground, so it would make some sense that a Jack Russell would have an interest in hunting rats and rodents, that was not what they were originally bred for. Rat terriers, however, were bred to catch rats and rodents, so if you are looking for a dog to do so, go with a Rat Terrier.
You, Snappy, are absolutely right. I saw a show on Animal Planet, dealing with that subject. They showed a man with 2 jack Russell's, who went "ratting". He brought them into a vermin infested barn, and they caught over 200 rats in a little over 2 hours, I believe. He said the closest rival they have for catching rodents, are cats. It was a very interesting show.
Jack Russell Terriers are now officially called Parson Russell Terriers. They were originally developed in England to hunt red foxes both above and below ground. For more information visit the AKC (American Kennel Club)'s website about Parson Russell Terriers. It has links to other resources related to this spunky breed!