I third the idea. Without interfaced stabilization, I wouldn't agree to quilt it, either. Unfortunately, starch may stabilize to sew the blocks together, but it wouldn't be enough to quilt them. Next time, I'd suggest using Pellon 906F iron-on interfacing. It's incredibly thin and stable, and adds no bulk to the shirts.

I just made a couch quilt (goes on the couch seats to protect from dog paws). I made the fatal mistake of using 'tapestry' fabric (which stretches and moves) along with thick poly batting (confession - a mattress protector). I 'tied' it using my machine, just spots of stitching in the one position (lockstitch?). Looks pretty good and less time consuming than hand tying.