Strange. My first thought was that the CNC4PC C37 has a latching relay, but that really doesn't make sense either. And from the pics on the website, they are using a diode to short back-EMF when the coil releases, so you can't drive it backwards anyway.
Just to double check, you have relay+ on the Zen box wired to IN on the C37, and relay- wired to GND (paired with IN, not the other one). Your spindle power supply (probably 12 volts) is connected to V_in. And the red wire on the spindle itself is connected to NO. Right?
I don't know the C37 board, but it looks like the 5V and GND pair are only used for the LEDs. If I'm right, you can leave those disconnected.
If you have everything disconnected from the C37, you should get continuity from V_IN to NO, and not from V_IN to NC, and not from NO to NC. Then hook up the relay+ and relay- from the Zen box to the IN and GND terminals on the C37 (+ to IN, - to GND). You should have continuity from V_IN to NC when you tell the software to turn the spindle on. When the spindle is off, you should have continuity from V_IN to NO again.