My problem was that I moved into a house where the previous owner had paired the GE toggle switches to his hub. Because they had previously been paired, they would not show up in the Ring app, even after performing the reset procedure (click up three times, down three times).
The only way I was able to unpair them was to buy a Samsung Smartthings hub, go to the Exclusion Mode setting in the SmartThings app, exclude the switch, and then it was available for pairing in Ring. At that point, I decided I liked the SmartThings hub better so I paired it with it instead.
The point is, Ring needs to add the Exclusion Mode workflow to their app, similar to the Exclusion Mode feature in the SmartThings app.