Draughts: King required?
No. You are never obliged directly to use a specific piece to move.

The only exception is trivial: a game played under forced capture rules where there is exactly one piece able to make a capturing move.

Even then, if that piece has a choice of captures, the player may choose which capture to make.

It is likely your opponent has either confused this rule with another, has played in environments with this uncommon variant rule, or the king had a capture available and no other pieces did.