We show that the collection of groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions is invariant under measure equivalence. In the case of orbit equivalence, the proof is mainly an application of a cocycle untwisting lemma of A. Furman and S. Popa, along with the fact that any orbit equivalence of free p.m.p. actions can be lifted to an orbit equivalence of the corresponding Bernoulli extensions of those actions.