Hm. A pure speculation would be that as it is probably a digital i/o, you could use either a pull-down resistor, or drive it with a logic gate (or equivalent circuit)...
The use of a logic level input would certainly be possible, however you would need to determine with a 'scope or multimeter, exactly what that level would need to be, and you would also probably be advised to use a diode to isolate your login input from the input generated by pressing the button (otherwise you risk frying your input device if someone presses the power button when your device is connected).
The power circuitry we know about uses a variant of the
Power manager, PWM controller MB39C303 From memory, having read the datasheet for this family a while back, this can be switched on with a fairly broad range of input voltages. This does not necessarily apply to cameras where we have no knowledge of the power supply schematics. These models may, or may not use a variant of this controller.
There is however a simpler method of switching on the camera, which works for all of the devices I have tested. The camera will power on, if the power button is held down mechanically, prior to switching on the external PSU. You can use a bit of tape, modelling clay or whatever to hold down the button, without any surgery being necessary on the camera.
Once powered on, the camera then behaves normally despite the button being continuously held down.
The problem then becomes a much simpler (and in many senses safer) one of controlling the external PSU. This can be done with a relay or mosfet, controlled by the external trigger (PIR, alarm clock, pressure pad, microcontroller or whatever).
