I would check the systems temp & fan sensors to see what they are telling me. I like this one but others will also work Temperature Gauge Pro. Paste in a screen shot so we can see whats up. Video editing can push the systems heat to the point the system shuts down as you describe here. Often systems need a good cleaning of dust from their fans and heat sinks but lets see what the sensors are telling us first.
When working on video or other heavy app I strongly recommend putting in more memory if you only have 4 GB. But before you do make sure your firmware is unto date follow this TN on how to check and update EFI & SMC firmware updates for Intel-based Macs. While Apple only support 8 GB you can in fact up your system to 16 GB. 16 GB maybe over kill for you if you are only working on short video segments.
Video editing can be tough on HD's. I would strongly recommend using an external FireWire or Thunderbolt connected drive (not USB) to host your work. As you often need to defrag the drive (between projects). If you don't have an external then it's likely your internal drive is highly fragmented. Over time they get this way and video/graphic editing pushes it. Before doing anything here make sure you have a good clean backup of your system and setup an external HD or flash drive as a bootable disk so you can boot up under it and then using Apples 'Disk Utility repair the disk & permissions. Then you'll need to run a defrag program, again there are a few of them out their I like this one and it offers other useful functions as well Drive Genius.
Lastly, make sure you have a good antivirus app and clean out the old log files.