Without seeing the PC first hand it is hard to tell. But I suspect you're probably stressing the PSU. Other issues that can cause reboots are faulty drivers and bad RAM.
Oh yeah is the computer dusty? Buildups of dust can raise the temperature of your PC, causing it to overheat and do strange things (especially reboot). So that is another aspect to consider.
Beyond ensuring a clean PC, the next cheapest option is to update drivers. All drivers -- USB, Videocard, Network/LAN/Wifi, Audio, motherboard, etc.
After that, THEN I'd start considering the hardware. First the PSU, then the RAM, followed by Videocard. If all of those check out fine then there is the hard drive, the motherboard, the CPU. in that order. CPUs rarely go so long as they are not overheating (and you're not overclocking).
You can buy a decent PSU checker for around 35 bucks (I prefer/use this one
Amazon.com: Thermaltake Dr. Power II Automated Power Supply Tester Oversized LCD for All Power Supplies - AC0015: Electronics).
Or you can just buy a decent new PSU for around 60 or 70 bucks.
Unless you have multiple computers, I'd probably just opt for the new PSU. Weight matters. Heavy coils are a plus. A lightweight PSU generally means a poor PSU. A poor PSU will be the bane of your computing experience. (I like Corsair CX PSUs -- using them in a handful of PCs)
For what you use, minimum 450 Watts. Looking at the 500w range wouldn't be a terrible idea. Anything too much over that is overkill imo.
That is only, though, if you spring for a new PSU. Check the other stuff first.
ITX is so tiny. You know you can upgrade to a larger case and put your ITX motherboard in it (then be able to upgrade to more powerful PSUs and videocards right?
Also, if you haven't already, you can disable the automatic rebooting if due to blue screen so you can actually SEE the blue screen of death. That way, it is easier to see what caused the BSOD (since it will show at the top of the screen.) I think the default is just to reboot (skipping the BSOD screen).
Hope you get it sorted