NRG greenboard is 98% air and therefore only 2% polystyrene
NOTHING can be mounted TO this. Your reel may be mounted THROUGH it to the studs behind.
From the Greenboard handbook:
17.1.5 Solid Blocking of Fitting and Accessories
Consideration should be given to the installation of
wall mounted accessories i.e. taps, electrical fittings,
etc. It is important to allow for adequate backblocking for these items prior to the
installation of the NRG Greenboard™.
Since this didn't happen, you have to add the blocking EXTERNALLY.
The typical 50' hose full of water is at least 15 pounds, the reel another 10 pounds and the pulling on the hose another 25 pounds or more, so you need a sturdy solution.
You would do well to distribute this across 2 studs, perhaps using 2 2x4 treated lumber. The screws would need to be long enough to go through the boards, outer covering (I hope the house isn't only covered in greenboard), the green board and at least 1" into the stud. Weather proof decking screws (ceramic coated for rust) would be a sturdy choice and come in 3", 3 1/2" and 4" lengths. All screw holes should be pilot drilled and filled with an exterior caulk or an exterior silicone.
Once the support boards are mounted to the studs, the reel can be attached to the boards with the original hardware.