No, there is no such species as a Red-necked Swan, and no, they aren’t from rural hilltops in Appalachia — these are Trumpeter Swans that apparently have been feeding in water rich in iron salts which have become deposited in their white head and neck feathers.
Way back in 1955, E.O. Hohn reported this discoloration of white feathers in a variety of waterfowl (e.g., Snow Geese, another all-white bird). He was curious whether this was some aberrant pigmentation so he leached the feathers with a mild acid, which removed the color entirely. He then tested the leaching solution for iron, which proved positive.
And there you have it — rusty-colored Trumpeter Swans marked by where they have been spending their winter in high iron-content water.
NOTE added after posting: Based on the amount of reddish-brown staining of neck feathers all the way down to the breast, I might speculate even further that these particular birds were feeding in deep, iron-rich waters, where they had to “tip-up” to feed (as shown below).