How to track reboots in Jelly Bean
System Tuner looks like it could help you figure out what's going on. Excerpts from the app's description:


  
  Display log(cat) for all/one process (even on JellyBean Android 4.1)  
  Record (optionally at boot) activities in the background  
  Record all apps or all processes  
  Allows analyzing past recordings without limits
  


I didn't try it for this special purpose, but the description suggests you can check what was going on when the device went down even afterwards: which processes were running, memory and CPU status, and more. As it is able to read system logs fine even on JellyBean, it should be able to give you the information needed to figure out what is going on.