The map $\pi_1(U) \to \pi_1(X)$ is surjective by Lemmas \ref{lemma-same-etale-extensions} and \ref{lemma-functoriality-galois-surjective}. \medskip\noindent Write $X^{sh} = \Spec(A^{sh})$. Let $Y \to X$ be a finite \'etale morphism. Then $Y^{sh} = Y \times_X X^{sh} \to X^{sh}$ is a finite \'etale morphism. Since $A^{sh}$ is strictly henselian we see that $Y^{sh}$ is isomorphic to a disjoint union of copies of $X^{sh}$. Thus the same is true for $Y \times_X U^{sh}$. It follows that the composition $\pi_1(U^{sh}) \to \pi_1(U) \to \pi_1(X)$ is trivial, see Lemma \ref{lemma-composition-trivial}. \medskip\noindent To finish the proof, it suffices according to Lemma \ref{lemma-functoriality-galois-ses} to show the following: Given a finite \'etale morphism $V \to U$ such that $V \times_U U^{sh}$ is a disjoint union of copies of $U^{sh}$, we can find a finite \'etale morphism $Y \to X$ with $V \cong Y \times_X U$ over $U$. The assumption implies that there exists a finite \'etale morphism $Y^{sh} \to X^{sh}$ and an isomorphism $V \times_U U^{sh} \cong Y^{sh} \times_{X^{sh}} U^{sh}$. Consider the following diagram $$ \xymatrix{ U \ar[d] & U^{sh} \ar[d] \ar[l] & U^{sh} \times_U U^{sh} \ar[d] \ar@<1ex>[l] \ar@<-1ex>[l] & U^{sh} \times_U U^{sh} \times_U U^{sh} \ar[d] \ar@<1ex>[l] \ar[l] \ar@<-1ex>[l] \\ X & X^{sh} \ar[l] & X^{sh} \times_X X^{sh} \ar@<1ex>[l] \ar@<-1ex>[l] & X^{sh} \times_X X^{sh} \times_X X^{sh} \ar@<1ex>[l] \ar[l] \ar@<-1ex>[l] } $$ Since $U \subset X$ is quasi-compact by assumption, all the downward arrows are quasi-compact open immersions. Let $\xi \in X^{sh} \times_X X^{sh}$ be a point not in $U^{sh} \times_U U^{sh}$. Then $\xi$ lies over the closed point $x^{sh}$ of $X^{sh}$. Consider the local ring homomorphism $$ A^{sh} = \mathcal{O}_{X^{sh}, x^{sh}} \to \mathcal{O}_{X^{sh} \times_X X^{sh}, \xi} $$ determined by the first projection $X^{sh} \times_X X^{sh}$. This is a filtered colimit of local homomorphisms which are localizations \'etale ring maps. Since $A^{sh}$ is strictly henselian, we conclude that it is an isomorphism. Since this holds for every $\xi$ in the complement it follows there are no specializations among these points and hence every such $\xi$ is a closed point (you can also prove this directly). As the local ring at $\xi$ is isomorphic to $A^{sh}$, it is strictly henselian and has connected punctured spectrum. Similarly for points $\xi$ of $X^{sh} \times_X X^{sh} \times_X X^{sh}$ not in $U^{sh} \times_U U^{sh} \times_U U^{sh}$. It follows from Lemma \ref{lemma-retrocompact-dense-open-connected-at-infinity-closed} that pullback along the vertical arrows induce fully faithful functors on the categories of finite \'etale schemes. Thus the canonical descent datum on $V \times_U U^{sh}$ relative to the fpqc covering $\{U^{sh} \to U\}$ translates into a descent datum for $Y^{sh}$ relative to the fpqc covering $\{X^{sh} \to X\}$. Since $Y^{sh} \to X^{sh}$ is finite hence affine, this descent datum is effective (Descent, Lemma \ref{descent-lemma-affine}). Thus we get an affine morphism $Y \to X$ and an isomorphism $Y \times_X X^{sh} \to Y^{sh}$ compatible with descent data. By fully faithfulness of descent data (as in Descent, Lemma \ref{descent-lemma-refine-coverings-fully-faithful}) we get an isomorphism $V \to U \times_X Y$. Finally, $Y \to X$ is finite \'etale as $Y^{sh} \to X^{sh}$ is, see Descent, Lemmas \ref{descent-lemma-descending-property-etale} and \ref{descent-lemma-descending-property-finite}.