I don't know if this is the correct way, but this is how I did it.
- you can mirror the mesh, then you have two eyeballs, make sure the mirror point is in center between the the two eyes, you may need to use an empty or the head of the mesh where the eyes are going.
- they eyes are part of rest of the armature, so you just symetrise those bones when you do the arms and legs, now you have the mesh and armature, make sure you name the bones correctly for this or it won't work. All on one side .l or .r
- if you mirrored the meshes it has the shape keys so you just need to set the drivers up like you did for the one eye. I used the same bones for both pupils, irises, which works if you want to make sure they stay the. same size.
- if you are using eevee for this make sure you set the blend mode in the materials setting so the cornea is transparent.
good luck, this was a hard tutorial for me when I did it.
Wayne has this course on rigging https://cgcookie.com/lesson/fundamentals-of-rigging-intro-chat which explains things really well.
