How is your puffer doing?
I see what you are concerned about. It isn't the black spots or the spikes that are naturally part of your fish, but the bright white things attached to your fish. It looks like your fish has either flukes, isopods or copepods. All are a parasites that attach to the outside of the fish.
You have to be very careful in treating puffers with medication. Puffers are sensitive to some meds, much more so then other fish. I'd recommend reading about treating puffers specifically before you attempt it.