I use one of those. One thing to keep in mind is it takes one press to turn it on and another to turn it off. So your rule would need to have an On followed by a delay of 1 sec. and then Off to simulate a 1 sec. momentary action. To clarify: you press the button once and the garage door opens, and another press will do nothing. You press a third time and the door will close, if you had let it go all the way open, then it takes a 4th press to get it back to the right state to open it again on the first press.
Press 1 - door opens
Press 2 - nothing happens
Press 3 - door closes
Press 4 - nothing happens
Press 5 - door opens again, same as press 1 and etc.
Not sure how handy or comfortable you are with doing this but I recently redid my whole garage door setup and love it. I had a myq setup and it was just a pain. No only does it rely on cloud connectivity, the thing would always drop off the network and according to their support they claim it doesn't work well with mesh networks. So I pulled it all out.
I did something like in the video below. Picked up a relay, and used a inovelli outdoor two plug zwave outlet to control the doors independently. I also soldered the relay directly to some extra remotes which allowed me to have the setup located inside the house by my rack and also allowed me to get around the myq "lockout" which prevents some of these other solutions from working.
The door status sensors I use are the xiaomi tilt sensors. I feel like I'm one of the few that have luck with these but any contact or tilt sensor can be used really.
The plus is that this solution is all local (zwave, wifi or zigbee depending on what switch you want to use). Its cheaper than these other solutions and you can add as many doors as you want. Works with my alexa/siri (through hubitat integration) with no limits. Doesn't have that annoying sound when closing (though I may add it back just a shorter version later).
The cons is that it requires a bit of work to throw together but not really a lot and creating rules etc can be confusing. I ended up just writing a basic app for my garage doors.
In the end I'm really happy I did it. I was looking at the gocontrol but when pricing it out I needed to get two units which was starting to run me $250. This was much cheaper.
If you are going to use a battery powered contact sensor be careful if you are in an area where it gets cold in the winter. Garage doors are notoriously drafty. Having a battery operated device basically directly in the path of sub-zero wind blowing in will have a big impact on performance of the device. I had to use an AC powered solution in the end. All the contact sensors I tried just wouldn't work. So, I use the Hubduino project to wire a sensor in. Also, you can use a relay with that board to control your garage door. There is a prefab driver that integrates the two devices at the board level. If you're a DIY and have any experience with Arduino it might be something to consider.
No, if you use 120, it will fry it. They're usually 12v DC systems. Not 120AC. Also, the controller board is sending out 12vDC and looking for that signal to be grounded. Supplying more current into the system isn't going to do anything. It has all the current it needs. When you press the button, you are making contact between two wires. That is what you need to simulate. And that's why you need the relay.