The ZD encoder is an "active high" device -- apply 5v to an input to trigger the related output.
If you look at the underside of the ZD PCB, all the wires/traces on the outside are tied together so that's the 5v "daisy-chain" and the inner wires connect to the inputs.
It sounds like your LED joystick is designed to work with a normal "active low" encoder, but it's impossible to give wiring advice without some images and/or details about
exactly what kind of joystick you have and if the LED wiring is connected to the microswitch wiring. (i.e. LED ground tied to joystick switch daisy-chain)
Scott