This could possibly be related to some sort of seizure activity. I would definitely have this puppy checked over by a canine neurologist. It would be important in any breed, but especially in a Staff because they grow up to be such large, powerful animals. At this point, you have no idea what it is that makes him squeal and bite. In a large dog, this could be very dangerous behavior. Please let us know what the vet says.
Ghilly
You say 7 weeks old, it that the age of the dog or the amount of time you have owned him? In either case, where did you get the dog? Go back to the source and seek advice. If a breeder that person should take some "ownership" in the problem, or at least be a resource of experience. If from a pet shop (a puppy mill possibility) I'd also discuss with the pet shop owner, not a hired hand.
Does the dog squeal constantly, or just when provoked?
What are you feeding the dog? Be sure to use a quality food, not most of the stuff at the super market. I consider Tractor Supply's 4 Health brand of kibble and canned to be moderate/low priced and top quality.
Use no-grain kibble and be sure to include some canned and make sure there is plenty of water. Don't feed scraps off the table or human left overs. Don't give any treats other than dog food that is selected per above guideline.
Sorry for you and the pup, hope these ideas lead you to an affordable solution. Good quality and limited variety dog may help.
If you don't feel you are getting anywhere with the one vet, it may be worth driving to a larger town or even a university with a neurologist. The dog could have a condition of the brain/spinal cord/peripheral nerves that are making him hyperaesthetic, but it could be more simple like a dermatologic problem..