Bower was born John William Kiszkan into a Ukrainian-Canadian family in Prince Albert, Saskatchewan, to Johnny Kiszkan, a labourer born Dmytro Kiszkan, and his wife, Lizzie, née Jacobson. (His father had previously been a homesteader.) He had one brother and seven sisters. He taught himself how to play hockey, using a branch as a stick, and made himself goalie pads out of old mattresses. Around age 10, his parents separated. An uncommon practice at the time, it created a stigma around the family in their town. When he was 15, he lied about his age and enlisted in the Canadian Army during World War II; from 1940 to 1943 he was stationed in England as a gunner with the 2nd Canadian Division. He was discharged due to rheumatoid arthritis in his hands.