It does not appear that these two John’s are the same. I have no answer as to the John/Conrad connection. Appears to be the same individual.

Suggest you search the Census records. Ancestry.com is a good place to go, unfortunately, I no longer have access.

As you can see, there are a number of variant spellings of the name.

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John Kountz, Private/Private, Company D, 1st Special Battalion (Wheat’s) Louisiana Infantry Reference Envelope, Cards filed with John Kuntz

John Kuntz, Wheat’s Louisiana, was last recorded present December 31, 1861 at Camp Carondelet, Va., there are no subsequent Company D muster rolls in the records

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Conrad Kutz, Corporal/Sergeant, Company C, 20th Louisiana Inf. (Confederate.), Reference Envelope, Cards filed with
John Kuntz

……

John Conrad Kuntz

Card #1 John Kuntz, Corporal/Sergeant, Company C, 20th Louisiana Infantry, 10 Card Numbers listed

Card #2 Jno.Kuntz, Corpl., Company C, Lovell Regiment Louisiana,* Enlisted Dec. 21, 1861 at Camp Lewis near

New Orleans, La. by Jno. G. Herman, 1st lt. L. Arty., A. A. A. G., for the war, present December 21, 1861

Card #3 Jno. Kuntz, Corpl., Company C, 20th Regiment Louisiana Infantry, present on January 1, 1862

Card #4 John Kuntz, present at the Jan/Feb, 1862 muster

Card #5 John Kuntz, Sgt., promoted Sgt., Mar. 1, 1862, present May 3, 1862

Card #6 Conrad Kuntz, 3rd Sergeant, Company C, enlisted December 21, 1861 at Camp Lewis by Lt. Herman,

present at the May/June, 1862 muster

Card #7 Conrad Kuntz, sick at Chattanooga since Aug. 20, 1862, absent at October 31, 1862 muster

Card #8 Conrad Kuntz, sick at Chattanooga since Aug. 20, 1862, absent at Nov/Dec, 1862 muster

Card #9 Conrad Kuntz, sick at Chattanooga since Aug. 20, 1862, absent at Jan/Feb, 1863 muster, last paid July 1, 1862

Card #10 Conrad Kuntz, absent at the Mar/Apr, 1863 muster, Absent sick, on furlough at Osyka, La. for 3 month since
April 7 last, last paid December 31, 1862

Card #11 Conrad Kuntz, October 31, 1863, Absent without leave since July 7, 1863 (Dropped)

Company C Record of Events
Received orders . . . to fall back to Jackson [Miss.] where we arrived July 7, 1863, and took our position in the trenches. Skirmished with the enemy until the 17th July. Evacuated Jackson on the 17th . . . .

Card #12 J. Kunz, Sgt. Co. C, 20 Regt. La., paid Sept. 22, 1862 $34.00 for the period July 1 to Aug. 31, 1862

No further records

* This company subsequently became Company C, 20th Regiment Louisiana Infantry

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Fred Kums, Private/Corporal, Company C, 20th Louisiana Infantry, Reference Envelope. Cards filed with
Frederick Kuntz

…..

Frederic Kooetz, Private/Corporal, Company C, 20th Louisiana Infantry, Reference Envelope. Cards filed with
Frederick Kuntz

……

Frederick/Fred Kuntz, Private/Corporal, Company C, 20th Louisiana Infantry, enlisted December 21, 1861 at Camp Lewis, near New Orleans by Jno. Herman, 1st Lt. La. Arty. for the period ending Sept. 7, 1862, missing since the Battle of Chickamauga, Ga., September 20, 1863, where captured, forwarded to Military Prison, Louisville, Ky. for exchange October 1, enlisted in Federal Army October 14, 1863, description: resident of New Orleans, La., light complexion, light hair, blue eyes, 5’ 8”, Conscript, enlisted in 11th Kentucky Cavalry

M320: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Louisiana

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Frederick Kuhn/Coon, age 22, Private, New Company I, 11th Regiment Kentucky Cavalry, U. S. A., mustered in October 5, 1863 at Louisville, Ky. for 3 years, appointed 7th Corporal, October 20, 1863, transferred to Company C, 12th Regiment Kentucky Cavalry, mustered out at Louisville August 23, 1865

This company was consolidated with Co. C, 12th Regiment Kentucky Cavalry July 2, 1865.

M397: Compiled Service Records of Volunteer Union Soldiers Who Served in Organizations from the State of Kentucky