Hi Bowey,
The output channels of the board should be able to connect to any Bpod digital input (BNC, Wire or the 'S' channel on the port interface board).
The bigger issue is how to power the breakout board. It requires 3-5V for power. The power supply must be grounded to the breakout board, and also to the input channel's ground pin (for BNC channels, this is the outer sheath). The state machines don't officially supply power in this voltage range (though if you want to get hacky, you could technically solder wires to the Arduino Due's 3.3V or 5V supply pins from above). You could also use a DC converter (make sure you find one that has good supply regulation) or a separate Arduino board.
If you're going to use a separate Arduino, you could also have it read in the lines, and relay events to the state machine. If you have Teensy 3.2 + the Bpod Teensy Shield, or Adafruit Metro M4 + the Arduino shield, you can wire the breakout board's output channels to any digital pin. If you have State machine r2 and you're connected to module channel 1-3, the state machine will power the Arduino board so you won't need a separate power supply. You'd then need to write a simple Arduino sketch to read in the logic, and return events (examples here). This is my preferred option because you'd be able to write filtering logic (e.g. to debounce the output channels), and generate events only for what you care about (e.g. touched = high).
-Josh
