MAIDUGURI, Nigeria—A 14-year-old girl in a black veil walked onto the road, raised her hands and made a plea to the soldiers nearby. She had been strapped to a suicide bomb and didn’t want to die.

It was explosives specialist Randy Iljeseni’s turn to defuse the girl, one in an army of child bombers deployed across northeast Nigeria by jihadist group Boko Haram. He grabbed a pair of fabric scissors, mouthed a brief prayer and slowed his breathing. He began to slice into her bomb-strapped belt. Some of his closest friends had...