You don't need an H-Bridge to drive this unit.
If you read the specs that the seller gives it will answer most of your questions -
Electrical - six wires exit the unit through the rubber grommet and have the following functions:-
Yellow - drives motor to pull shaft in (with 12 volts on Yellow and earth on Orange)
use a transistor or MOSFET controlled by 1 Arduino output to retract the shaft
Orange - drives motor to push shaft out (with 12 volts on Orange and earth on Yellow)
use a transistor or MOSFET controlled by 1 Arduino output to extend the shaft
Blue - activates electromechanical clutch
use a transistor or MOSFET controlled by 1 Arduino output to activate the clutch
Green - earth- or ground
Red) -Probably one end of the position feadback pot Try +5 V here
White) - either end of resistance wiper circuit - probably the wiper of the feedback pot. To an Arduino Analog Input.
NOW - if you were to replace the internal circuitry with an H-Bridge you would still need the transistor or MOSFET to control the clutch. It would be much simpler to use a couple transistors/MOSFETS to control it than investing in an H-Bridge.