There are many examples.

For example, in the case of uncountable models, let the language have one unary predicate $U$, and let the predicate be satisfied in ${\cal M}_1$ on a countable set with an uncountable complement, but vice versa in ${\cal M}_2$, with the predicate holding on an uncountable set with a countable complement. These are both models of the theory of an infinite co-infinite predicate, which is complete because all countable models are isomorphic, and so they are elementarily equivalent. But there is no elementary embedding in either direction, since this would involve mapping an uncountable set into a countable set.

But you said you wanted a countable example. Let us modify it by adding infinitely many constant symbols $c_n$ for $n\in\mathbb{N}$. Let us have $U(c_{2n})$ and $\neg U(c_{2n+1})$ in both models, as well as $c_n\neq c_m$, and furthermore each model has a unique $x$ that is not the interpretation of any constant symbol $c_n$. In ${\cal M}_1$, we have $U(x)$, but in ${\cal M}_2$, we have $\neg U(x)$ for this object $x$. The models are elementarily equivalent, since the corresponding theory in any finite restriction of the language is $\omega$-categorical, but there is no elementary embedding in either direction, since there is no place to send the additional object $x$.