You mention the unit restarts. I take it the complaint means it isn't fully restarting and is restarting immediately after login. At what stage does it not finish restarting?
Is the data backed up?
The reinstalling of the software option is probably the best one to try first in event you have some improper background extension that is causing the machine to restart after login. You can try to login in safe mode, by holding the shift key.
You can also see if you can get yourself an external hard drive and use command-R to install the OS on the external drive, or command-option-shift-R. Be sure the external drive is wiped or gets a new partition that is HFS extended or APFS and at least 70GB using Disk Utility. From there once installed you can see if you can see the internal hard drive and copy over critical data before reinstalling the OS on the internal drive.
Once your data is backed up, you can wipe and restore the OS and see if the issue persists. This can at least eliminate any hardware issues you have with your computer. If the computer will not install an OS fresh on an external drive even with internet restore (command-option-shift-R), that's when going to the authorized service provider is advisable. Shop around. http://www.apple.com/buy some providers offer options to try to replace the battery before replacing the logicboard. That's a much cheaper option if AppleCare Plus can't be extended.
From the sounds, you may have something that is overheating the computer. If you can't extend AppleCare, and the Exchange Repair program does not offer a cheaper means of resolving he problem, contact local Apple Authorized Service providers. They may be able to diagnose whether or not the battery has overheated and needs replacing
