Do not 'join' the eyes to the head.  Keep them as separate objects for rigging, and parent them to an appropriate head bone, keeping the offset.   My Auto-Rig Pro tutorial provides more detail, and the concepts work for any rig.

Both eyes may be using the same shader or geometry nodes.  When you duplicate the eyes for left and right, make sure you also create duplicates of the eye shaders and geometry nodes, then append the names with something like ".L" and ".R" 

In the Modifier Properties panel, move the Geometry Node modifier above/before the Rig modifier.

The eyeballs are procedural, so you need to view it using the 'Material Preview' tab of Viewport Shading.  The eyelids are not procedural, but they will be difficult to see unless you use the material preview mode.