[Phlogiston]

If you look at the back of the counter circuit board, there are two push buttons at the left. The lower switch is the one you are interested in. 

There are four terminals on the PCB switch. You want to connect one wire from your external switch to the lower right terminal on the PCB switch. The other wire from your switch connects to power supply ground, so when you press your external switch you are momentarily grounding the lower right terminal of the existing switch.

Mine does not have the buttons. I'll look to see if the lands are there.