Question
I am in the process of finishing my basement and have a lot
of scrap drywall. I was wandering if it would be ok to put
it on my garden and till in to help break down the clay?

Answer
Hi Ken, I would not recommend it. Even if you were 100% sure that there are no potential toxins in it from the manufacturing process, it would be difficult to get it ground up and uniformly distributed through the soil.  Plus, it will change you pH, so you'll have to monitor it and may have to adjust it frequently. I'd much rather use organic matter to break up the clay.
Jim