The equilibrium constant for the reaction A ⇌ B is 4 at a certain temperature. What is the equilibrium constant for the reverse reaction B ⇌ A at the same temperature?

Reversing a reaction inverts its equilibrium expression, so $K_{reverse} = \dfrac{1}{K_{forward}} = \dfrac{1}{4} = 0.25$. An equilibrium constant is never negative, and squaring is what happens when the equation is doubled, not reversed.