Here's part of the version that J. W. Washington, Fort Myers,
Florida, contributed to Louis Chappell, who published it in his 1933 book (pp 116-117).
They carried John Henry down the smoky road
And put him on that long white road.
When they brought that poor boy back to town
He was lying on his cooling board.
suggesting that "white road" could be the original that mutated, in several versions of "John Henry," to "White House." I have suggested that the limestone around Leeds and Dunnavant, Alabama, might account in some way for the whiteness of a road there, perhaps surfaced in marble chips.
There is a much simpler explanation. Many version of "John Henry" state that "they buried him in the sand." Very close to the east portal of Oak Mountain Tunnel, Dunnavant, where John Henry is supposed to have beat a steam drill in 1887, is Sand Ridge Cemetery. The area has exposed sand on several of its southwest-to-northeast trending ridges. Surely Sand Ridge is named for this. There was, and is, a ridge road leading to that cemetery, and if it were sandy in 1887, then it would have been "white." Unfortunately, the inventoried markers at Sand Ridge Cemetery don't include one for John Henry. On the other hand, the inventory is known to be incomplete, and the cemetery contains many graves for which markers have not been found. It may be that the inventory is based on a cursory search. I intend to go over there soon and have a personal look.
At least one version of "John Henry" states that he is buried by a river. Many versions imply that his burial is within sight of the railroad. Also very close to the alleged site of the contest, the railroad crosses Shoal Creek, which meanders through a flood plain of sand. This spot will be easy to examine because the highway is very close to the track at that point.