I found the 1860 census reports on White and Gilham, but I found nothing about their military records in what little I have here. Perhaps somebody else can help you with that.

Both "Wm. K. White" and "Hugh Gillam" are shown in the same page 423 of Grand Pass Township in northeast Saline County.

William K. White is shown as 25-year-old, Virginia-born, farmer of modest means ($600 worth of real property and $500 in personal property). He lived with his wife Mrs. Mariah White, age 27, born in MO. In the same household lived 35-year-old, Virginia-born, carpenter Allen McNeal ($600 private property).

Gilham is three households from White in the Calvin Alezer household of Calvin, wife, and small child. Alezar is a miller by trade. Three other men in their twenties lived in this household. One is a lawyer. Hugh Gilliam is 28, born in MO, and also a miller by occupation. Nearby live a blacksmith, laborers, and such occupations that make me wonder if these all live in a village, perhaps centered on the mill.

Sorry I couldn't help you more.

Bruce Nichols