Toggle state between two circuits with two momentary switches
What you need to build is called a Bistable Multivibrator. Basically with a pulse as applied, the circuit will toggle from one state to the other.

With multivibrator circuits, you can have two output - one being the inversion of the other. With such an arrangement, you could then use something like an optocoupler to connect each output to one of the "black box" circuits.

Interestingly, this is exactly what a flip-flop is. If you get a D-type flip-flop with Q and !Q outputs, all you need to do is wire the D input to the !Q output, and connect your push button to the clock input. Finally you connect the Q an d !Q outputs to the two optocouplers which are then connected to your boards.

To improve things, you'll also need some debouncing on the push button to avoid multiple clock pulses. This can be done using an R-C circuit and a Schmitt trigger to generate a nice clock pulse from the push button.

I can draw up a circuit if that is easier to follow.