Schreier's father was a textile merchant from Gösting whose business collapsed in the 1920s; after the Nazi occupation of Austria, the Schreiers tried to flee abroad and Bernhard was taken to Palestine with one of his sisters in 1939 (two others had permits to travel to the United Kingdom); their parents later joined them in Palestine. During the Second World War, he volunteered in the British Army; after the formation of the Israeli Defence Forces in 1948, he was commissioned into the new Engineering Corps and served with them in the Arab–Israeli War that year. He subsequently became a civilian mechanical engineer, working on roads and infrastructure, eventually establishing his own contracting business.