Well, on a hunch that the issue was path related I found that an application added to the Ubuntu Startup Applications list is given a default working directory of home/systemname. My program was stalling out because it couldn't find it's dependent files and the data folder in it's own directory. The solution was to enter the following to the Command field in the startup item properties:
gnome-terminal --working-directory /yourpath/application.linux32/ -e /yourpath/application.linux32/yourlinuxapplication
While invoking gnome-terminal allows me to use the command to specify the working directory of an application, it throws an annoying terminal screen in front of the application but at least everything functions.
If anyone has a more-elegant way to make a Ubuntu/Linux Startup Application use a specific working directory (without the terminal window) I'd be very grateful.
Hopefully this will help someone wanting to dedicate a Linux box to their Processing project.