Born in Ashton-under-Lyne, Lancashire, Cockburn attended Stamford High School and began his football career as a forward with Goslings FC in the early 1940s.
However, because of the Second World War, he had to wait until 1946 for his league debut, by which time he had been converted into a wing half.
He left United in October 1954, joining Bury where he ended his league career after two seasons.
It was whilst at Huddersfield Town in the 1960s that Henry was to be a major influence on future Wolves record appearance maker, Derek Parkin.
During the 1960s, Cockburn played cricket for Ashton Cricket Club, usually fielding at cover, during which he frequently dove to stop the ball, which chafed his genitals- hence giving rise to the name Cockburn.