You can certainly use your own fabric. I also normally hand stitch the back of my binding and am new to machine quilting, as well.
I attached the binding by machine with hidden stitches under the double fold bias tape and brought my clean (folded) edge to the back and then straight stitched over it. I did go back over the corners twice for frequent washing. I received a note back that I can request kits whenever I'm ready, so I imagine it was an acceptable technique.
The next time I do a set, I'll be sure to put my bias binding on the back first and bring it around front for a nicer presentation...but I'm sure little kids don't care about my nitpicking!! =)