That sounds like a third-party program running in the background stealing the game's focus and crashing it in the process when you're triggering the Print Screen key.
Check your systray (lower right on the desktop) or task manager for anything in that regard. You also might want to use CCleaner or some similar tool to check the Windows startup entries of automatically run programs you don't want or need (though that's more advanced stuff).
MWO features a built-in screenshot mechanism utilizing Print Screen, which on key press saves a JPG of the actual screen into your Windows user profile under the path "Saved Games\MechWarrior Online\ScreenShots", so there's no need for any additional software.
Small amendment: actually Print Screen AND F12 work, so try F12 as it might not trigger your troublemaker.
Edited by Ace Kirby, 04 May 2015 - 11:12 AM.