Born in Wauchope, New South Wales, Everingham was educated at Fort Street High School[2] and graduated with a Bachelor of Medicine and Surgery from the University of Sydney in 1946 and worked in public and private hospitals and as a family doctor.

[4] Local Rockhampton Labor identity, Barry Large (former Member for Capricornia Kirsten Livermore's long-serving political advisor), recalled that Everingham placed anti-smoking stickers on cigarette vending machines throughout Parliament House in Canberra due to his anti-tobacco stance.

[4][7][9] Everingham was one of many Queensland Labor MPs to lose their seats at the 1975 election held after the dismissal of the Whitlam government, defeated by the National Country Party's Colin Carige.

[4] His death prompted a number of tributes including from long-serving state Labor MP Robert Schwarten who described Everingham as the "father of Medicare" due to his campaign for Australia to have a universal health care system.

The car collided head-on with a truck on the Warrego Highway near Laidley west of Brisbane, while Stephen and his passenger were returning to the Queensland Agricultural College at Lawes where they were both students.