Bob Lubarsky and I have been working on this during the current academic year, and today we submitted it for publication. (Update: the paper has been accepted to the Archive for Mathematical Logic.) You can read a preprint here: ElementaryEpimorphisms.

Abstract

We show that every $\Pi_1$-elementary epimorphism between models of $ZF$ is an isomorphism. On the other hand, nonisomorphic $\Sigma_1$-elementary epimorphisms between models of $ZF$ can be constructed, as can fully elementary epimorphisms between models of $ZFC^-$. Elementary epimorphisms were introduced by Philipp Rothmaler.. A surjective homomorphism $f: M \to N$ between two model-theoretic structures is an elementary epimorphism if and only if every
formula with parameters satisfied by $N$ is satisfied in $M$ using a preimage of those parameters.