Even simpler. No need to unsolder the button the board or add wires to the board. When you press S2, it pulls the signal "PHONE_ON_N" low. The signal PHONE_ON_N is connected to pin 4 on the low-speed connector, you can add an external switch between Pin 4 and pin 2 (GND). The switch should be normally open, and when you press the switch it shorts pins 2 and 4, which is exactly the same as pressing the power-button on the board.
The power button signal is part of the 96Boards CE specification. Many of the available mezzanine boards have a power-button on them because when you install a mezzanine board you can no longer reach the on-board buttons. Same with the Reset Button.
Full Disclosure: I am an employee of Qualcomm Canada, any opinions expressed in this or any other post may not reflect the opinions of my employer.