Usually a rule of thumb, what a dog can contract viral or bacteria, Humans do not share the same...
Humans catch intestinal bacteria thru different sources than an animal...so I do not think at all your dog is why the children are going thru an episode of intestinal flu or bacteria...
Worms in dogs for example: you should look up on internet the different worms that can exist in a dog that has been exposed....but, for a human to contract the same worms would take a little more than sleeping in the same bed...unless, the dog is infested w/worms of what kind yet??? And, beleive me if your dog was infested w/ worms, your dog would show signs of laying around, lethargic, not eating properly, either eating too much or not eating at all, the dogs belly could be swollen, and the dog eating normal, but the dog too skinny, the dogs feces show little rice looking pellets in the stool, and the dog scrathing tail across the ground, (but, just because a dog scratches butt against the ground does not mean there is worms, dogs need to scratch butt across grounds to keep anal glands extracted, which is normal dog behavorial....
Has your dog been taking the routine meds and immuzations by your vet? If so, you could probably rule the dogs out...
A lot of kids are getting this running bowels w/slick greenish shinny watery type stools, with a terrible stinch smell, odor you won't forget, last about 3 days...I just got slip of the mind, I can't remember the name of the problem...
Keep us posted....excuse the long comment...
I don't think so, but MAYBE worms could be passed on if the boys handle the dog's feces and then put their hands into their mouths. Have you noticed any worm pieces in the dog's stool? Does the dog do the butt-scootin' boogie on the floor to indicate he has a problem with his rear end? If not, he likely doesn't have worms. Most heartworm preventives like Sentinel also take care of intestinal parasites as well. If you don't regularly clean your yard of dog poo, start doing so now. I have a flat bladed garden shovel that I use for that purpose, and with 2 medium sized dogs, I have to do poo patrol every day. I certainly don't want to step in it, and don't want the dogs dragging it into the house if they step in it either. :-)
You probably should have your dog examined by a vet just to make certain there are no problems, and also take your boys to the doctor. Even if the boys didn't pick up something from the dog, diarrhea for over a week should be investigated by your pediatrician.