From the joysticks I've dismantled in my time autofire simply closed the joy button circuit... No special electronics at all.
Autofire generally uses a 555 timer. I'm not sure what joysticks you've taken apart, but they don't sound great. Unless the auto fire toggles on and off then it isn't auto fire & you can't do that with just a switch.
555's work best with higher voltages, so it might get upset with 3.3v
If might work, you'll have to test it.