If this were an issue with UberStudent, it seems it would happen no matter the power source. It makes no sense at all that what you described happens while on battery but not
when plugged in to the wall with the power pack. I'm concerned that your new laptop has a hardware issue.
Do you have any other operating systems on your machine? Does the same thing happen with them? Since this is a new laptop, you might wish to investigate this right away in case it is indeed a hardware issue and is covered under warranty. Even if you need to install something, say, vanilla Xubuntu, you might wish to test this.
So that would be my first suggestion: Test your machine with another operating system and see if the same thing happens. If it does, you've found the problem.
It also does not seem to be an issue with the software that monitors your battery supply. If it were prematurely thinking your battery was at end-of-power, your machine would suspend and not reboot - which, again, makes me suspect a sudden power supply interruption due to a hardware issue.
I don't think it has anything to do with the proprietary Broadcom driver, but in the meantime please do the following:
Open up a terminal (hit F12 on your keyboard in UberStudent) and type in and run the following:
dpkg -s bcmwl-kernel-source
The above command queries whether or not the package bcmwl-kernel-source
is installed. Read the result from running the command and it will tell you. Let me know if bcmwl-kernel-source is installed or not.
Cool-looking desktop, by the way.