It is correct to say, "John is lying on his back." The other sentence is incorrect grammar. Lay (lay, laid, laying)is a transitive verb meaning “to cause to lie, to put in a recumbent position, or to put or set down’’ as “lay the child on the bed,’’ “lay the box on the table.’’ Lie (lie, lay, lain) is an intransitive verb meaning “to be in or place oneself in a recumbent position; rest; recline.” Lay (to put, place or prepare) and lie (to recline or be situated) are often confused in their application. In the senses mentioned, Lay always takes a direct object, lie never does. He laid the newspaper on the desk. The hen laid an egg. The carpenter lay the foundation for the house. He often lies down after lunch. I was lying in bed when he called. China lies in Asia. Parents used to teach the distinction to children by saying, “Hens lay eggs, people lie down.” The widespread misuse of "lay" in the US is probably because our schoolteachers are unaware of the distinction.

They are both correct. John is laying back in his bed-describes what John is doing-John is lying in bed, he is not sitting in a chair. John is lying on his back on his bed-describes the position john is in-John is lying down, he is not sitting up.