SEOUL—After years working in North Korean coal mines, two former prisoners of war are seeking nearly $300,000 in damages from an unlikely source: North Korean leader Kim Jong Un.

Han Jae-bok, 85 years old, and Ro Sa-hong, 90, were captured behind enemy lines during the 1950-1953 Korean War and, after fighting was halted by an armistice, were put to work alongside other South Korean prisoners of war. They spent 33 months in forced labor and decades more consigned to work at the coal mines until they escaped to South Korea....