I've discovered the problem and have found the solution (See Below).
The Problem
A Windows directory needs to exist:
C:\Users\USERNAME\appdata\LocalLow\Sun
On your box, you have the following directory structure, most likely:
/home/USERNAME/.wine/drive_c/users/USERNAME/
*Where USERNAME is your actual user name
The Solution:
Make the subdirectories
(Point and Click Method)
Open Home Folder
Open .wine
Open drive_c
Open users
Create Folder appdata
Open appdata
Create Folder LocalLow
Open LocalLow
Create Folder Sun
(Console/Terminal)
cd /home/`whoami`/.wine/drive_c/users/`whoami`/
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 :)
Importance/Relevance
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.
Reference