Why is my computer shutting down when under pressure?

"I would have come to the same conclusion as you have. I would have thought that this was either related to power or overheating.

It is worth going in to System Settings > Advanced (not giving exact instructions as not sure if Windows 7/Vista/XP), and then under startup and recovery choose ""Disable automatic restart on system failure"" as this will rule out you getting BSOD's without knowing.

I often find that socket 775 fans are very hard to know if you have installed correctly. I personally put a flat head screwdriver in each of the four slots and strike the top of my screw driver with a hammer.

As soon as it shuts down, go in to the BIOS and take a look at temperatures / health settings and see if you can see anything unusual - typically if it is above 70, it most likely is that the heatsink is not on correctly.

This sort of issue is not easy to diagnose in this method without seeing the computer, but hopefully this has helped you."

"Based on the comment conversation with Wil, it sounds like the issue may be with proper installation of the CPU cooler. Sites like AnandTech and TomsHardware have extensive reviews for CPU cooling solutions ranging from cheap to ridiculous.

Some other things to note:

Make sure all 4 screws/pins of the CPU cooler are firmly locked in place (as mentioend by Wil). I personally wouldn't use a hammer, but I've found that it often requires some firm pressure with a thumb. You'll feel it snap into place when you've done it right. A slight tug backwards should tell you if you did it right.
Get a utility that tells you the CPU temperature while your computer is running. I personally use the one that came with my motherboard, but it looks like EVGA doesn't provide one for your mobo (from what I could tell). Kevin K's suggestion is probably a good one (no experience here). An alternative is RealTemp. If the temperature is more than 65 to 70 deg C, then overheating is your problem. Right now, mine is 32 deg C (I'm in a cold room).
Make sure you don't use too much thermal compound. The purpose of the compound is to fill in microscopic cracks and microscopically uneven surfaces. The compound itself is less conductive than either of the two surfaces, so if you've got too much, it will actually make things worse. Follow the directions that came with the thermal compound or will come with your new heatsink (since you said you broke a retention pin on the original).
Your new power supply has two 12V rails. Essentially this means that it has two independent 12V power supplies. Some other power supplies have up to four rails. The 12V rails are often the most important ones these days because they are used to power the CPU, graphics card, and hard drives. Make sure you balance the current drain on the two rails. Look at the power supply documentation to find out how you can tell which connectors are on which rails. For your system, I'd put the CPU on one rail, and the graphics card and hard drive on the other.

If you ever decide to add more powerful graphics cards and/or add several hard drives, you'll eventually need to upgrade the power supply. Use the pro version of the eXtreme Power Supply Calculator to figure out how many amps you need for 3.3V, 5V, and total for 12V. Note that the total watts of a power supply is a near-useless number (even though everyone tries to use it). Make sure you have enough wattage overall. Although the linked site isn't free ($1-5), it's very cheap and you'll easily make up the cost in your electric bill (if it helps you choose a less-powerful supply than you would have otherwise selected) or headaches and freeze-ups (if it helps you avoid one with too little wattage on at least one rail).

Did you plug in a power connector directly onto the graphics card? Based on the newegg picture, it appears to need a 6-pin PCIe power connection (look at the top right corner when facing the fan)."