It appears that you are experiencing the classic symptoms for the overheating issue described here: http://www.intel.com/support/motherboards/desktop/sb/CS-034326.htm. You can request a thermal pad kit for your Intel NUC that should take care of the problem.
I neglected to mention that the unit was powered down and removed from AC for 12 hours. It was cold when re-tested this morning.
Unless there is some thermal protection disallowing a restart, it would appear to be another problem.
Since the unit will not boot, I an unable to make adjustment to the fan speed. If there is a protection method preventing boot, I do not see how to restore it in that article.
It sounds like you do have a defective unit that needs to be replaced. The quickest way to get a replacement would be through your place of purchase. If that is not possible, you can send an email to Intel Customer Support (http://www.intel.com/support/mailform/desktop/nuc/emailsupport.htm) for further assistance.
When requesting warranty replacement, you will need to include the following information:
Serial number and SA Number (found on a barcode label on the bottom of the chassis)
Also, please also reference this community thread for the warranty agent to review.
Here's an update with a new unit. On first boot Windows 7 Setup saw the SSD but reported it as 0 size and 0 free. Reboot showed BIOS reporting the drive there.
I reseated the SSD but after that it never showed back up in the BIOS.
The heat thingy is on the inside cover on this unit where it wasn't on the 1st. There's also a black piece of tape that I'm not sure what to do with.
I flashed the BIOS to 46 and cleared the BIOS but SSD still will not show. I'm wondering if this device may be a root cause of all the problems.
Is there anything else that can be done or is it time to assume the SSD is bad?
Yes, I agree with your conclusion that it's not the NUC, but the SSD that is likely defective. The black tape you see is supposed to be wrapped around the wireless antennas inside the chassis. You can just throw it out - it doesn't need to stay.