So we now have a machine running Ubuntu 12.04 and we did some testing with the xbox 360 controller. I got it working using the XBoxDriver tutorial on this site: http://www.omgubuntu.co.uk/2013/07/dealing-with-xbox-controllers-in-ubuntu
After following the instructions, I tested the game and confirmed that the buttons do not autofire. Double jump seems to work fine.
The problems people are having might stem from having multiple drivers running. I know xpad comes standard in Ubuntu and I had to blacklist it. That might account for multiple keypresses registered with one push.
Respond here with the distro you're using and the drivers you're using to get the gamepad working if you're still having problems with this.