I haven't done a trackball hack and have never worked with an arcade trackball beyond looking at Happs. What I can say is that on my Atari encoder boards that fit my steering wheels, there are 5 wires on the mouse (2 for transmitter, 3 for receiver) for each axis that connected to 4 wires for each wheel (1 axis).
The mouse wires are +5v, ground (for transmitter), +5v, ground (for receiver) and signal. I think one of the wires on the Atari board is not connected, so that means you need 3 wires for one axis or 4 wires for 2 axis.