Is your script for setting the skin in Awake() or Start() on a gameobject? Or is your menu enabling the gameobjects with the SkeletonGraphic and setting its skin immediately after that?
I believe the Skeleton only becomes created/accessible after Awake() has been called on the SkeletonGraphic.
My guess with it working in-editor but not in your build is that the Script Execution Order happens to work when running it in the Editor, but isn't the same on the build. AKA if you 'got lucky' that Awake() is being called on your SkeletonGraphics before your SetSkin code when running it in the Editor. But I could be wrong.
If you want to set the skin before activating the gameobject, then I believe you'd need to let the gameobject be active when you're loading up your scene - like, let it call Awake(), and then set it to disabled, that way it has the chance to initialize.
