I also veto the idea of using spray baste on the squares. As delicate as the fabric is, looks like batiste, the spray may damage and stain the blocks.
I agree, I'm not so sure I would quilt it either, BUT, if you must, please use muslin as this would fit the time period of the fabric and redwork. Back in those days, the popular quilting pattern would be cross hatch. Also, I would only use 100% cotton thread with it. YLI puts out a nice cotton thread for quilting.