I would check the bobbin area for a piece of thread caught somewhere.

Just tried this. The beginning stitches look like regular old stitches, nothing special. Looking at the fabric, I have a nice long thread at the front and another in the back, they meet in a nice stitch and it continues in the same way.

Thanks homespun. I've just cleaned the bobbin area with extra attention. I also changed my needle once again (I don't buy expensive ones so I don't mind discarding them). I have to press 30 pieces before I can try again with my quilt but I did a small chained sample and it worked ok. I think I've incorporated most of your bits of advice. Obviously, new needle and super clean bobbin/feed dog area. When stitching I am going very slowly and carefully between the 2 squares and I'm also keeping the squares much closer together so there is less time with no fabric under the needle. So far so good.... The proof will be after my pressing. I'll report back. Thanks for all your help!!

Well, something worked wonders! I've just finished chain piecing 30 pieces with zero thread breakage. I wish I knew exactly which remedy was the one that worked but I'm glad I'm up and running again. Thanks to all who helped!!

It could be something as simple as . . . old thread!

Thanks!! Honestly the number of possible reasons for the breakage frustrate me. I'd like to know the exact reason things happen with my machine and often I don't figure it out, the problem just goes away and I never know what fixed it. In this case I admit that I was not using a high quality thread, but it was a new spool, not old stuff. After I realized changing the thread didn't solve the problem I went back to my original thread and still had the problem. But something else fixed it because now the thread isn't breaking anymore and I'm still using my original thread. Ugh.

I chain piece on an old Singer from the 1920's. Just before I slide the next fabric under the presser foot, I lift the foot and slide the previous piece about 1/4", lower the foot onto the next piece, and away I go. It gives room for the scissors and less chance of the seam opening at beginning and end. I have had that crazy "cat's in the cradle" thing happen when my needle needs changing.

I don't leave but a couple of stitches between pieces when I chain stitch. I never have thread breakage. I have found some thread spools have thin pieces of thread on them or other flaws, get past those and will sew much better. Never use to have that problem with any thread, finding those areas more and more. Never affects thread in the bobbin.
When changing bobbins I always try to jiggle the bobbin to make sure it is in there correctly. My machines have side mount bobbins and a small space.

Wonderful news!

Not sure why the thread is breaking but you should butt them up against each other, you will use a lot less thread, and less tails, and no extra trimming.