Sorry - I wasn't auto-following the thread, so those two replies that didn't mention me didn't catch my attention.
A box around something in Daz Studio usually means that it knows there's an object there, but it doesn't have the geometry for it. I'm wondering if, when you exported as OBJ, the eyes actually got included in the export. If they're separate objects in Blender, it depends on your Blender settings whether everything in the scene is exported, or whether only the object you have selected is going to be exported. That might be why they didn't transfer.
To rig the character, you're going to have to create the bone structure, and then "weight map" all the surfaces to the bones so that the mesh moves with the bones. Daz Studio has a "transfer utility" that will allow you to apply rigging semi-automatically which may help, but for that to work properly, you're probably going to have to scale the figure so that it's approximately the same size (and in the same pose) as one of the Genesis figures. If you're going to want to be able to move the ears, you'll have to create new bones for them and weight-map the ear surfaces to them manually.