Download the VBox from the kali site: Kali-Linux-2016.2-vbox-amd64
Loaded into VirtualBox, everything works. Take a snapshot in case updating breaks kali or something else.

1) apt-get update = Works fine, finishes, no problems.
2) apt-get install virtualbox-dkms = Works fine, finishes, no problems.
3) apt-get upgrade = Goes through the rather long process of downloading and updating everything. Gets to a part where it reboots the VM, and then takes me to the Kali login screen. I use the default username/pass since I didn't change them, but every time I enter the password and press enter, the whole VM restarts and goes back to the login screen again. Never changes. Had to revert to my initial snapshot of the machine.

There were no error messages. Only thing that works to get me back to a working Kali version is to revert to the initial snapshot. Doing an apt-get dist-upgrade causes the same problems. Rather frustrating problem, since I can't update Kali. Don't know if it's a problem with a new kernel compiling and breaking Vbox, or something else. If it were the former, I would definitely assume the dev's know about this and would have thought there would be more threads about this sort of issue.