Dave -
I had a very clean SQ-Ai body recently that was exhibiting similar issues. The battery check light was erratic, even with different sets of fresh batteries, and the shutter would often fail to fire. This was consistent even with different lenses, backs and finders attached. I also tried cleaning the contacts and pressing on the battery door, but to no avail. Luckily, I had just bought it as a backup to my SQ-A body, which has worked flawlessly for over 12 years. I assumed there was some internal electronic issue, so the seller refunded my money. Unfortunately, I can't help much as I never determined the real problem, but did you recently buy this body?
That being said, I'm much more confident with the SQ-A and will probably buy a second one soon. I believe there were some electronic changes with the SQ-Ai; different battery type, motor drive capability, off-the-film plane metering with flash, etc.
Good Luck!