Hi,
From what I'm hearing, it could be a overheating issue. Since X-Plane is a very demanding application, it can wreck your PC. I used to have a similar issue to yours given the situation you described. My PC was getting a black screen just after 3-5 min after launching the game, then It would proceed to reboot. Either you need a better CPU cooler/cooling or a new PSU.
Download Core-Temp and check your temps if they are going crazy.
I bought the Cooler Master Hyper 212 Evo CPU cooler and a new PSU and that solved it.
You could list specs but try that first
-Rishi