Alright... I found how to do this.
If you want to run any of your own custom scripts once Knoppix finishes booting up, just place them in /home/knoppix/Desktop directory. Then from console, run "saveconfig" and a GUI dialog should pop up asking you which settings and files you want to save. Check 'Desktop' and uncheck everything else and save the files on some drive. This will generate a 'knoppix.sh' file and a 'configs.tbz' file, which contains all user-defined settings and scripts. Copy these files over to any portable medium you have (most likely a USB drive). Edit 'knoppix.sh' to include the following line at the end to run your script once Knoppix finishes booting up:
cp /home/knoppix/Desktop/myScript.sh /
That code just copies your script over to / directory and then executes it. Use the following command as a boot-up option to ensure Knoppix searches for a 'knoppix.sh' file to execute when booting up:
The reason why my script wasn't booting off a USB drive because it wasn't mounted when Knoppix is booting up. If I try to mount the USB drive and then run the script, Knoppix simply assigns a different device name to the USB drive.
Hope this helps out anyone looking to accomplish something similar.