I forgot to mention a couple of ideas that have helped us and others:
1. Using a consistent phrase to tell your dog you want him to pee/poop works quite well. Our breeder uses the words "hurry hurry". When we got Kipawa, I started doing that with him in the backyard and it worked - but it takes time, patience, a few treats and TONS of love and praise when he does it right. Kipawa does tend to not want to pee in his back yard, but has no problem pooping. Funny little character!
2. Just recently we bought Poochie Bells. We hang them on the back patio door. Kipawa has started to figure out that if he rings those bells we are right there to open the door for him.
I also wonder if it is a confidence issue for some basenjis (that they want their peeps with them when they pee). Maybe they feel vulnerable when they are out there doing their business. Kipawa and I have a tight bond, and I know he feels more comfortable if I am there or at least if I am watching him from the kitchen window.
Hope some of these ideas help.