Flowers was born in Jena, Louisiana on November 18, 1943. He was named after his grandfathers Woodie and Claude. His father, Abe Flowers, was a welder and inventor. His mother, Bertie Graham, was an elementary-school and special education teacher. As a boy, he showed mechanical aptitude like his father, Abe, and he earned the rank of Eagle Scout. When he was seventeen, he and four friends were driving on Louisiana Highway 127 when they were hit head-on by another vehicle that was traveling at about . The accident killed two people in Flowers' vehicle and one in the other. It made him believe what he called "genetic opposition to violence" and created his "fierce, vocal loathing of any spectacle that involves crashing pieces of machinery into each other with deliberate force."