I looked at William Guilliams service record which says "Died at Frozen Rock" as opposed to "killed" which may indicate he died from sickness or may have been wounded elsewhere and later died at Frozen Rock. The time of his death, July 25, 1862, was during the 1st Indian Expedition so there may have been Union scouts on the north (east) side of the Arkansas in this area. The Confederates were on the south (west) side the Arkansas with headquarters at Ft Davis (just north of Bacone College) and scouting up the west side of the Verdigris River.
What can you tell us about William Guilliams? Was his last name really McGuilliams (or McGilliams or McWilliams)? There is a Wiley McGuillaims in the same company. Both enlisted in Bryan's Battalion at Evansville AR (co. A?) which became company H of Adair's 2nd Cherokee Mtd Vols.
Ken