Individual Details

Henry doesn't appear in a census with his parents - seems to have been omitted. I did not know he existed until his death certificate showed up on FamilySearch.org.
Montana Death Certificate, Registers No. 171
Henry Harmon died at the Veterans Administration Facility at Fort Harrison, Lewis & Clark County, Montana on 5 Sep 1938. His residence was Chinook (Blaine Co) Montana. He was Single and his birth listed as 2 Aug 1858 in Zionville, Indiana. He died of pulmonary edema, with arteriosclerosis and myocarditis. Parents listed as William A. Harmon and Elizabeth Miller, both born in Indiana. The informant was H. C. Watts of Ft Harrison. His body was removed to the Crow Agency, Montana on Sep 8th.

He was buried Custer Natl Cemetery, Crow Agency, Big Horn, Montana.

This hospital is still open today.

I did find an enlistment record:
1881
Henry Harmon, March 31, at Indianapolis by Lt. Randall. His home was Boone Co, IN and he was age 22, a Farmer. He had Grey eyes, Black hair, dark complexion, was 5' 9 1/2" tall.
18th Infantry, Company D
Discharged March 30, 1886. Service at Fort Hays, Kansas. Private. Character Good.
Apparently this would have been for the Apache Wars.
The Apache Wars were a series of armed conflicts between the United States Army and various Apache nations fought in the southwest between 1849 and 1886, though minor hostilities continued until as late as 1924.

Master Index Card, Veterans' Administration
Henry Harmon, Private, Company D, 18th US Infantry
At Cleveland, Mont.
Born 2 Aug 1858; died 5 Sep 1938
Enlisted 31 Mar 1881; Discharged 30 Mar 1886

He has been very elusive in any census record. I have wondered if he had been married at some time.

Birth	2 Aug 1858	Zionsville, Boone County, Indiana
Death	5 Sep 1938	Fort Harrison, Lewis and Clark County, Montana

Father	WILLIAM ALEXANDER HARMON (1826 - 1881)
Mother	EMMA ELIZABETH MILLER (1825 - 1871)
Sibling	James N. "Taylor" Harmon (1848 - 1933)
Sibling	MARY ELIZABETH "Molly" HARMON (1849 - 1912)
Sibling	John A. Harmon (1852 - 1881)
Sibling	Adora Harmon (1853 - 1854)
Sibling	Charley M. HARMON (1856 - 1859)
Sibling	William Harmon (1860 - 1933)
Sibling	Minnie B. Harmon (1866 - 1946)