The OPR birth indexes on SP show only one John Muirhead born in Hamilton in 1850 to William Muirhead and Agness (sic) Jackson.
The 1861 Scottish census likewise shows only one John Muirhead, birthplace Hamilton. The household (living at 50 Surrey St, Govan, Glasgow) is as follows:
William Muirhead, age 50, carter, birthplace Kilbride, Lanarkshire
Agnes Muirhead, age 38, wife, birthplace Hamilton
William, age 16, son, victualler, birthplace Hamilton
Mary, age 14, daughter, dressmaker, birthplace Hamilton
Thomas, age 13, son, shop boy, birthplace Hamilton
John, age 11, son, scholar, birthplace Hamilton
Agnes, age 8, daughter, scholar, birthplace New Kilpatrick, Dumbarton
Margaret, age 6, daughter, birthplace New Kilpatrick, Dumbarton
George, age 4, son, birthplace New Kilpatrick, Dumbarton
Cheers,
Rodeo