Sounds like a severe system stability issue. Real-time video encoding is a VERY heavy computational load, and can cause many normally light to medium-duty usage systems to overheat almost instantly when the demands shoot up.
Are you monitoring system thermals? What are your temperatures on average, and after starting stream?
Are you overclocked at all?
Did you write down the BSOD error code and Google it? They contain information on WHY the system just crashed, after all. You may also be able to use BlueScreenView to pull up the most recent one.
Have you done a memory load and burn-in test with something like Memtest86+ for a few hours (at least 2-3 complete cycles, more being better)?
Are you monitoring your voltage levels? If everything peaks at once on a system running with a questionable or inadequate PSU, the ripple/vcore dip can be enough to disrupt a stable CPU voltage and pull a BSOD crash or freeze.
OBS wouldn't be the root cause of these. There's another underlying problem with the machine that needs fixing.
