Ahh, the couple ways we can do this. You actually had 2 questions from what I see.
Question 1: How do I launch a bash script at boot?
Answer 1a: As root, make an upstart launch script , see upstart docs.
Answer 1b: Option 2 as user with no root access. Create a cron job that runs every 5 minutes to check to make sure bash script isn't already running and if it isn't, start said process.
Question 2: How do I launch the application in VNC/RDP?
Answer 2: RDP is not well supported. VNC is not very secure, but if you insist. Start vnc server once by hand, set up password. This will also create the .vnc directory for the configuration. Now edit the .vnc/xstartup and add the startup of the program/bash script.
Going back to Answer 1a+b, you should then add the startup of vncserver as your startup and not the script itself. The starting of vncserver will then start the application once.
Example for cron:
*/5 * * * * pgrep -u $USER vncserver > /dev/null || vncserver -geometry 1024x768
Mileage of above script varies. Feel free to add a bit more verbosity.