Wow, Aimee, that looks like a great endeavor.
I would still suggest stay-stitching around the outer edges of each of the squares before you sew them together into your larger pieced sheet of background fabric. If you were to use a product like Fray-check or Fray-stop you might find the stiffness it causes to be detrimental to your applique later on.
Great idea thanks but the centre I will be appliquing over is 35 nine and a half inch blocks sewn together so I can't really square them up later. The stay stitch has merit though. Here is a link to the Block of the month so you can see what I am talking about. https://www.susanclaire.com/detail.jsp?Product=BMTCU
Hmm, that didn't highlight so you may need to paste it in.
I quite often do a stay stitch about a sixteenth to an eighth of an inch in from the outer edges of my background block when I do applique. The stay stitch I use is often shorter than my normal sewing stitch. I also cut that block at least a half-inch larger than what the finished size will be. For example, if the block finishes at 12-inches square, I will cut my block 12-3/4 inches or 13-inches square. That will take into account any fraying and shrinkage taken up by the applique.
I have a new block of the month that is made from Japanese woven taupes. I have to piece the background and then it will be appliqued. I am not worried about the applique just the background. Is there a good method apart from fray stop to stop it fraying?
