Here are three William Thompsons I found in units which were involved at Lone Jack. I could not locate a Robert Thompson.

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William Thompson, Company C, Shanks’ Regiment Missouri Cavalry*, enlisted August 14, 1862 in Jackson County by Col. Thompson for the war, promoted to 2nd Corporal February 20. 1863, present at the Jan/Feb, 1864 muster, surrendered and paroled May/June, 1865, resident of Jackson County, Missouri, no further records

* This regiment appears to have been variously known on the field as 2nd Regiment Missouri Cavalry, Jeans’ Regiment Missouri Cavalry, and Shanks’ Regiment Missouri Cavalry; but it was designated by the Confederate War Department as the 12th Regiment Missouri Cavalry

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W. H. Thompson, 1st Lieutenant, Company I, 3rd Regiment Infantry, 1st Division, Missouri State Guard, served from August 1, 1861 to January 8, 1862, no other records

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William J. Thompson, 1st Lieutenant & Adjutant, Field & Staff, 6th Missouri Cavalry [Hooper’s Regiment], surrendered, paroled at Shreveport, La. June 15, 1865, resident of Lewis County, Missouri

M322: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Missouri