You can use the machine quilting thread to piece, but using a regular 100% cotton thread works just as well. Whatever you do, never mix up the machine quilting thread with the hand quilting thread. The two are not compatible!!! I learned the hard way and had to take my machine in for a good cleaning. The two threads are wound differently and the hand quilting thread is waxed, which is bad for the machine.
Boo, what a sad thing to have happen! I'm told the polyester thread they now make isn't supposed to do that, but I won't use it because of the very problem you encountered. Polyester thread is much stronger than cotton and as the quilt ages the thread will rub against the cotton fabric, actually cutting the fabric over time. I took a class from Linda Jenkins once and she was telling us about how she and her partner took a tour of the thread manufacturing plant they use. The man in charge told them to never mix polyester thread with cotton, and for that same reason. I know many quilters who don't seem to have a problem with it, but I would be devastated if a quilt I spend hundreds of hours on ended up falling apart because I used the wrong thread. You have my complete sympathy!