Any time I hear someone send a dog off for training, I get concerned. Dogs are not the ones who need training, the owners are. The problem is not the dog. The problem is the owner needs to learn to train. Boarding trainers are scams who make my blood boil almost as much as charlatans like John Edwards. Almost without exception, if you can't learn to train your pet, you don't need to own it. If you are not willing to put the effort into learning how, you don't need it. And trainers worth their salt know that, yet prey on people instead of being honest. Please, get your dog home and get some help.
The S in basenji stands for stubborn. HELLO. No trainer will stop that. You train around it. Unless you put a shock collar on the dog and zapped it into a seizure, few basenjis give up being stubborn. It is part of their charm. You learn to train smart so they think what you want them to do is a good idea. Please, bring your dog home. You have no idea what damage they are doing to him.
TRAINING isn't about tricks or performance. It is about building a relationship. YOUR relationship with YOUR dog. The problems you describe are, I am sorry to say, typical. Really. If I tell Arwen to get off the kitchen table once in a week, I tell her 5 or 6 times. Yes, I could booby trap it so she gets scared, but really… is that want I want? Especially since, being a basenji, she'll simply find another place I don't want her to go. A big part of the UNappeal of the breed is the need to watch the little sucker actively. Many here have dogs they can leave uncrated or unwatched. Most of us, not so much. We had a lovely foster here who didn't even bother TRASH with kleenex in it. My daughter begged to keep him! But most, you just learn to be better house keepers.
If basic normal basenji behaviors are not what you want to live with, or are willing to live with in order to have one... you might want to consider contacting the breeder about rehoming him and find a dog more suited to your needs. But if you do want to keep him, you are going to have to meet his normal behaviors about 80 percent of the way, because that's just what they are.