You need to troubleshoot all the standard causes for breaking threads.

Check to see that your thread size is compatible with your needle size. It's possible your thread is too big for the needle.
Make sure the thread is threaded correctly - unthread and re-thread carefully, paying close attention to the manual's instructions.
Make sure the thread path is not impeded in any way.
Change to a new needle.

Did you use a spray starch or basting glue on your quilt sandwich? How much did you use of the spray starch/glue? If so, could be the thread is catching and creating unwanted tension which is breaking your thread. Check to make sure the bar that rides close to the needle is not bend (sorry but I am not near my machine to check the manual for the correct terminology). If bent, replace. It is imperative not to use a heavier weight thread then the manual suggests. Another thought is that your quilt sandwich may be pulling off the table which would cause added tension on the thread creating breakage. Hope this little bit of info is helpful!

I found a dealer close to home and told her about my problems. She said to bring it in and she would trouble shoot it for me. She worked on it over 2 hours and the breakage happens at all different times and doing different things. She told me it was no longer under warranty which I'd owned it less than 6 months. I went home and tried to find something on that but found nothing in writing as to how long the warranty was. Had I been offered the gold warranty which I guess costs you another grand it would have been. She told me it would cost me to ship it back to the company, cost me to have them look at it, cost me to get it shipped back. She told me that they had an embroidery machine from babylock that was fussy like mine and paid to send it back to be told that it was just fussy and to deal with it. She told me I would be throwing my money away to send it back, that I just had a fussy machine and to deal with it. She said go home, sew with it and when the thread breaks , rethread it and move on. That was back in December and it is still in the box from bringing it back from her place. I haven't touched it. We did everything possible to see if that was the problem. The one thing that I notice is that when the thread breaks you can hear it snap. This to me says its getting caught.

The engineering of these machines interests me and I like watching them work, but I would never use it enough to justify even 10% of the price. I don't like how the stitches on the back don't match the top, and I don't love the hand-stitched look enough to pay for a machine to mimic it.

Plus I'd have nowhere to put it. All my spare sewing machine space is full of vintage machines! LOL

......" A fussy machine and deal with it!"....... Not in my book.....not at that price! I would be sitting down and writing a scathing letter to the CEO ..certified mail...and Demand repair/ refund on product..return at no cost to you......if not a report to the better business bureau would be filed.........I would be steaming mad about the situation.......

I'm with Geri on this one, I'd be spitting nails! I'd be the biggest squeaky wheel, too. I'd be calling them every day, standing on the sidewalk outside their store with a sign, checking to see if I could file a complaint with Angie's List, etc. Not sure if the dealer is lazy or if she was telling the truth about the manufacturer, but either way it's bad customer service.

Back in Dec. I found a local quilt shop who works with the shop where I purchased it. She said to bring it in she would trouble shoot it. Turns out that it isn't under warranty. At that time I'd had it less than 6 mths. Could find nothing in the paperwork that told me what my warranty was. She asked if I had purchased the extended warranty which from what I have read is another grand. She played around with it for 2 hours and told me if I sent it in, I would have to pay shipping, pay them to check it out and pay return shipping. The thread breakage has no rhyme or reason to it. She said that some of babylock's machines are just touchy. She told me to take it home and use it and when the thread breaks to rethread it and keep going. It is still in the box from when I brought it from her shop home back before Christmas. I could care less about getting it out of the box.

I've done sashiko on the embroidery machine. Watched what it did. I can duplicate simple lines and curves with a regular machine and the bean (triple straight) stitch. I've done it on quilted sweatshirts. (Not premade sweats and converted) It was mostly border continuous stitching from a stencil. I used poly embroidery thread and some metallic.
You can put together straight line quilting designs very effectively for a block.

I had a couple of quilt people ask how I did it when I wear the shirt.

Are there any freelancers who will quilt my queen-size on a sashiko machine ?

My friend has one and you are right-it only makes one stitch but she does use it quite often and it is pricey. She uses it on all of her wall hangings.