My dog's back leg is paralyzed, no pain or feeling. What's wrong?

There are a couple of reasons that a rear leg will become paralyzed. The most common are a lack of blood supply or a lack of nerve supply. The lack of blood supply most commonly occurs because of a blood clot and is very painful, so I doubt that is what is going on with Nook. The other possibility is that the nerves that go to the leg are damaged. The most common reason this will occur without any signs of trauma or injury would be a bulging disc (which again is most commonly somewhat painful) or a condition such as a Fibrocartilaginous Embolus (which is like a stroke to the spinal cord). FCE's are non-painful, but there isn't any treatment other than time and physical therapy. Unfortunately, the only way to determine the cause is often an MRI. I'm not sure if you have a veterinary neurologist near you, but they might be able to help you with a diagnosis and treatment. If not, I would consider trying to find a veterinarian who does acupuncture or rehabilitation to help you manage and improve Nook while you are waiting to see how much he recovers.
I hope that was helpful. Thanks for using Petco Pet Education Center, formerly Petcoach.