You can make a chicken nugget from pure chicken breast meat, but they would be too pricey for fast food. Let's face it, deep frying anything already took the healthiness factor down 10 fold.
The muscle tissue, blood vessels, and skin, all probably come from mechanically de-boned chickens (MD), which is basically squeezing the extra pieces of meat off the bone after the breast and other primal cuts are cut out as too not waste any product. It's perfectly safe to eat, but looks much more disgusting than "pink slime" that the media had a field day over, and put companies out of business. If the wrong vegetarian media person saw MD, they would have it all over the news (kind of like this) on how bad it is for you. Even though it is still probably much leaner and better for you than a steak for instance. It's the deep frying it that makes it unhealthy.