The fog lights on my 04 X will only turn on when the headlights are on and turn off again if you turn your brights on. Is there a way to configure/rewire them so they can turn on and off independent of the headlights?

Fog lights are for fog. If you're in fog you lose visibility with your high-beams on, thus they don't work with the switching and only work in low-beams. Often times I find in bad fog, I don't even want my low-beams on, but just my yellow PIAA-fogs and the driver lights on.

If you want to change it, then you probably have to re-wire the power source for the relay.