Laptop autoshutdown problem?

Try monitoring your system with RealTemp or SpeedFan and see if you're having a temperature problem.

When I first saw the question title and first bit, I thought heat... but after reading what you wrote, I don't want to scare you, but, my first guess would be either hardware failure or just a driver issue.

It is worth a try making sure that all the vents are clear as @techie007 said just to rule out heat.... if your machine ever crashes in Ubuntu, then, this is the most likely reason.

Firstly drivers:

It is possible that you have a piece of hardware that is incompatible with the driver on the Windows installation media. If you are installing from a source that did not come with your machine, you may want to try another disk in case it was a restricted disk without (or with a invalid) storage controller, or similar that is causing a blue screen.

However, that being said, I have seen very similar issues a few times.

Most hardware have a few ways that they can be used. Firstly is "just working" and secondly is "full-feature rich". An example of this is a graphics card - with a generic driver (such as "Generic SVGA Driver") will work fine for displaying a picture, but, give it the full graphics driver and performance will go through the roof!

To expand further, on a laptop I had about 3-4 years ago with the dreaded Nvidia bug, it simply would blue screen and crash with the Nvidia driver, but, by uninstalling it and using the standard generic SVGA driver, it would work fine - and - just like you, I was able to use Linux just fine as the build I had did not have a good graphics driver.

Without seeing the laptop, it is hard to guess, but, based on my past experience and what you said, this is what I think the problem is.