Put your multimeter in ~10VDC mode, and hook the leads to the two red wires on any given switch you showed in your diagram. Reverse leads if your MM reads backwards and is not autopolarity. Switch not pushed, you should see a voltage; switch pushed, no voltage. Caution: if there is only 3.5uA through the switch, you must use a high input impedance (~10megΩ) Multimeter, an analog meter wont cut it.
Are you sure that the circuit in the box doesn't apply some sort of multiplexed AC voltage to the switches. If it has a uprocessor in it, it probably multiplexes the switches?