This is a major clean-up of my previous argument, thanks to the very helpful comments of Emil Jerabek.

I interpret the question as follows: Characterize all sentences in the language of fields that are preserved under field extensions.

Claim: A formula $\phi$ is preserved under field extensions if and only if the field axioms prove that $\phi$ is equivalent to an existential formula.

For definiteness, we take the language of fields to have operations for addition and multiplication, and distinguished elements for 0 and 1.

The main tool to establish the claim is the following

Lemma: Suppose $M$ is a structure in any old language, and $T$ is a theory in that language. Let $\forall(T)$ denote the set of universal consequences of $T$. If $M\models\forall(T)$ then $M$ extends to a model of $T$.

Proof-sketch: The diagram of $M$ cannot be inconsistent with $T$, because otherwise $T$ would imply the negation of some formula in the diagram of $M$, and therefore would imply the universal closure of the negation of that formula, contrary to the assumption $M\models\forall(T)$. But then any model of the diagram of $M$ plus $T$ will do the job we require.

Now we prove the Claim. The non-trivial part is to show that if the theory of fields (which we will henceforth call $T$) cannot prove the equivalence between $\phi$ and any existential sentence, then $\phi$ is not preserved under extensions.

But our premiss is equivalent to the statement that $T$ cannot prove the equivalence between $\neg\phi$ and any universal sentence. Let $S$ be the set of universal consequences of $T+\neg\phi$. Then $S+T\nvdash\neg\phi$. For otherwise, using the fact that $S$ is closed under conjunction (at least up to logical equivalence) we could argue that $\sigma+T\vdash\neg\phi$, for some sentence $\sigma\in S$. But this, together with $T+\neg\phi\vdash\sigma$ would imply that $T\vdash\sigma\longleftrightarrow \neg\phi$, contrary to the premiss of our argument.

We have shown that $S+T\nvdash\neg\phi$. Therefore there is a structure $k$ satisfying $S+T+\phi$. Evidently $k$ is a field. Recalling that $S$ is the set of universal consequences of $T+\neg\phi$, we can now use the Lemma to conclude that $k$ extends to a field $K$ satisfying $\neg\phi$. This proves that $\phi$ is not preserved under extensions.

I'd love some feedback on this!