What is this electrical box with a button?
It's how the city reads your water meter. You should be able to trace the wires back to your water meter, or if that isn't feasible, look at your water meter and you'll probably see a device on it with two wires of the same color.

I doubt that "button" can actually be pressed, rather it's the contact point where the meter reader places their probe to take the reading.


(source: thetimes-tribune.com)