GHIM hydraulic transom switch

1.) Confirm pin-outs of your new switch.  I would think one of your two center terminals is where your voltage would go.  Place one lead on a multimeter in the center, depress the switch, and put the other lead on the terminal in the direction you're pressing.  If you're in continuity mode, you should hear a beep.  If you're measuring resistance, the resistance should be 0.  Confirm the other side.

2.) Now, we need to figure out which is your voltage source.  I would suspect it is the red one.  Measure voltage between this wire and an available ground hopefully nearby.  If you get 12v, then put this one on one of the center terminals.  If you don't have a ground nearby, then you can probably just put the red one on first in the center.  If steps 3-5 don't work for you, then back up to this spot and try the brown one in the center (my second guess).  If I zoom in enough, I think that is a yellow/green wire, which may suggest ground.  Try testing for 12v between these two.  If that's the case, red on one of your center terminals and yellow/green on the other center terminal.

3.) Then, start with just one wire putting it on one end of the switch on the side that you put the red wire.  The white one for instance.  Activate the switch.

4.) If that worked, try the other wire on the other end on the same side that you put the red wire.  Activate the switch.

5.) If up is down and down is up, swap wire positions.

This is the diagram for the factory NIMBLE (read: Junk) antenna/switch.     I used it when I rewired mine to a GAMA switch.   Not sure if this is what you were looking for:

Sorry,   I thought you were looking for the switching from the stern or the wireless remote to ACTIVATE the hydraulics.   Not the hydraulics wiring itself.    I can't speak for the 588 or other boats, but on the 575 the switch on the stern as well as the wireless remote both led to the same junction box (in my picture) that activates the gate hydraulics.    When I replaced my remote I started with the wireless unit, which told me which wires were up / down and then replicated those connections for the switch in the cockpit.  

The schematic I posted is the wireless remote antenna connections.   In other words,  if you follow the antenna (on the 575 mounted on the inside of the hull in the starboard aft void) it should be wired like this diagram. 

If I have it wrong, I apologize.   Good luck.