Since $X$ is quasi-compact, there exist finitely many affine opens $U_i \subset U$ such that $U' = \coprod U_i \to X$ is surjective. After replacing $U$ by $U'$, we see that we may assume $U$ is affine. In particular $U \to X$ is separated (Schemes, Lemma \ref{schemes-lemma-affine-separated}). Then there exists an integer $d$ bounding the degree of the geometric fibres of $U \to X$ (see Morphisms, Lemma \ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}). We will prove the lemma by induction on $d$ for all quasi-compact and separated schemes $U$ mapping surjective and \'etale onto $X$. If $d = 1$, then $U = X$ and the result holds with $Y = U$. Assume $d > 1$. \medskip\noindent We apply Lemma \ref{lemma-quasi-finite-separated-quasi-affine} and we obtain a factorization $$ \xymatrix{ U \ar[rr]_j \ar[rd] & & Y \ar[ld]^\pi \\ & X } $$ with $\pi$ integral and $j$ a quasi-compact open immersion. We may and do assume that $j(U)$ is scheme theoretically dense in $Y$. Note that $$ U \times_X Y = U \amalg W $$ where the first summand is the image of $U \to U \times_X Y$ (which is closed by Schemes, Lemma \ref{schemes-lemma-semi-diagonal} and open because it is \'etale as a morphism between schemes \'etale over $Y$) and the second summand is the (open and closed) complement. The image $V \subset Y$ of $W$ is an open subscheme containing $Y \setminus U$. \medskip\noindent The \'etale morphism $W \to Y$ has geometric fibres of cardinality $< d$. Namely, this is clear for geometric points of $U \subset Y$ by inspection. Since $U \subset Y$ is dense, it holds for all geometric points of $Y$ for example by Lemma \ref{lemma-stratify-flat-fp-lqf-universally-bounded} (the degree of the fibres of a quasi-compact separated \'etale morphism does not go up under specialization). Thus we may apply the induction hypothesis to $W \to V$ and find a surjective integral morphism $Z \to V$ with $Z$ a scheme, which Zariski locally factors through $W$. Choose a factorization $Z \to Z' \to Y$ with $Z' \to Y$ integral and $Z \to Z'$ open immersion (Lemma \ref{lemma-quasi-finite-separated-quasi-affine}). After replacing $Z'$ by the scheme theoretic closure of $Z$ in $Z'$ we may assume that $Z$ is scheme theoretically dense in $Z'$. After doing this we have $Z' \times_Y V = Z$. Finally, let $T \subset Y$ be the induced reduced closed subscheme structure on $Y \setminus V$. Consider the morphism $$ Z' \amalg T \longrightarrow X $$ This is a surjective integral morphism by construction. Since $T \subset U$ it is clear that the morphism $T \to X$ factors through $U$. On the other hand, let $z \in Z'$ be a point. If $z \not \in Z$, then $z$ maps to a point of $Y \setminus V \subset U$ and we find a neighbourhood of $z$ on which the morphism factors through $U$. If $z \in Z$, then we have a neighbourhood $\Omega \subset Z$ which factors through $W \subset U \times_X Y$ and hence through $U$. This proves existence. \medskip\noindent Assume we have found $Y \to X$ integral and surjective which Zariski locally factors through $U$. Choose a finite affine open covering $Y = \bigcup V_j$ such that $V_j \to X$ factors through $U$. We can write $Y = \lim Y_i$ with $Y_i \to X$ finite and of finite presentation, see Limits, Lemma \ref{limits-lemma-integral-limit-finite-and-finite-presentation}. For large enough $i$ we can find affine opens $V_{i, j} \subset Y_i$ whose inverse image in $Y$ recovers $V_j$, see Limits, Lemma \ref{limits-lemma-descend-opens}. For even larger $i$ the morphisms $V_j \to U$ over $X$ come from morphisms $V_{i, j} \to U$ over $X$, see Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation}. This finishes the proof.