I've discovered the problem and have found the solution (See Below).
A Windows directory needs to exist:
On your box, you have the following directory structure, most likely:
*Replace USERNAME with your actual user name
Make the subdirectories
(Point and Click Method)
Open Home Folder
Create Folder appdata
Create Folder LocalLow
Create Folder Sun
mkdir -p appdata/LocalLow/Sun
*NOTE: appdata/LocalLow/Sun IS case sensitive, so type it exactly as is shown
Double click on the JRE executable (I used 32 bit, but I believe 64 bit works too. Be sure to check winehq's website and your version of Ubuntu to be sure)
Install away :)
As there are some programs that are Windows specific, and only partially use Java, I found that I still needed a solution to this problem. Therefore, I have provided the aforementioned procedure to solve the problem.