There is really no such thing as a "hypoallergenic" dog. All dogs shed hair, and dander (skin cells, these are usually the source of human allergic reaction), some more than others. Smooth-coated JRTs will shed more than the rough coated....in general, short haired smooth coated dogs shed at a fairly constant rate year-round (long-haired dogs shed their undercoats only in the spring). There is a product called "Allerpet" that works very effectively to prevent the shedding of hair & dander, and resulting allergic reactions.....fur stays on the dog until brushed off. You can pick it up at most pet stores & vet offices.
My apologies, I too am new to the site & did not see the extrapolation you posted with your question. Here's a full answer.... I can't resist pointing out that children raised with pets from infancy have been shown to develop much greater resistance to airborne allergens than other children (my baby-sitter was a German Shepherd, can't imagine a family without a dog). It's not the fur that's the problem....it's the dander, or shed skin cells. That said, unless your oldest son's allergic reactions are severe enough to pose significant health risks, by all means add a puppy to the family!
All dogs shed (even poodles). Some less than others. Wire coats the least, short smooth coats the most, though both shed at a fairly constant rate. Long-haired dogs shed their entire undercoats in the spring, but only for those few weeks. Allerpet works beautifully, but for any dog brushing every few days will be necessary. Hypoallergenic filters for your air & heating ducts are a great idea, and the Dyson "Animal" vacuum is a godsend. On the flip side, some of the long-acting prescription antihistamines can work wonders...I had a client with who bought a Pug, developed violent allergic reactions (we're talking anaphylactic shock), and the combination of Allerpet and Nasonex solved the problem completely.
CAVEAT: A Jack Russell Terrier is probably a bad idea for boys that young. They are extremely high-energy, need a great deal of exercise (as in running a few miles a day), can bark excessively, are stubborn as boxes of rocks, smart as whips, fearless, and too fast for you to catch. Terriers were made to kill badgers, weasels, gophers, and so have a strong prey drive, dominant personality, Napolean complexes and little patience. JRT's in particular were bred as horse companions, barn guards, and vermin killers, so are best suited for very active adult owners. I wouldn't trust one not to bite a 6 year old.
I AGREE - a Jack Russell is a very strong willed dog and once it sets it's mind to something, it is very difficult to redirect. I have a smooth coated JRT and she sheds like nothing I have ever seen. If anyone out there has a solution for both the shedding and getting that white barbed hair our of upholstery, I would love to hear it. I brush her daily and it doesn't seem to make a difference.