The following assumes that you have a relatively recent MiKTeX distribution. The same effect should be reproducible on older versions, but the interfaces may be different.
The above works fine as long as your computer knows where to look for the UCClatex directory. This is located on Euler, so if, when clicking on Computer from the Start Menu, you do not see the UCClatex or LATEX directory available then open up File Explorer and type \\euler into the address bar which will bring up the folders to which you have access. There should be one called LATEX; right-click on this folder and select the option Map network drive.... Choose a letter for this drive (assumed to be Z: above, but change it to whatever you like), and then go through the above instructions.
Back to the uccmathexam.sty page.