Let $j : \eta \to X$ denote the inclusion of the generic point into $X$. From the low degree terms of the Leray spectral sequence, we have an exact sequence $$0 \to H^1(X, j_\ast \mu_n) \to H^1(\eta, \mu_n) $$ and thus it remains to prove that $j_\ast \mu_n = \mu_n$. Let $U \to X$ be an affine etale open. Then since normality is local in the etale topology, $U$ is also normal. In particular, connectedness is equivalent to irreducibility and thus we may assume that $U$ itself is the spectrum of a normal domain $A$ with fraction field $K$.

Let $f\in K$ satisfy $f^n = 1$. Then $f$ is integral over $A$. But $A$ is integrally closed in its field of fractions and hence $f \in A$. This proves that any section of $j_\ast\mu_n$ comes from a section of $\mu_n$ and we are done.

Let $j : \eta \to X$ denote the inclusion of the generic point into $X$. From the low degree terms of the Leray spectral sequence, we have an exact sequence $$0 \to H^1(X, j_\ast \mu_n) \to H^1(\eta, \mu_n) $$ and thus it remains to prove that $j_\ast \mu_n = \mu_n$. Let $U \to X$ be an affine etale open. Then since normality is local in the etale topology, $U$ is also normal. In particular, connectedness is equivalent to irreducibility and thus we may assume that $U$ itself is the spectrum of a normal domain $A$ with fraction field $K$.

Let $f\in K$ satisfy $f^n = 1$. Then $f$ is integral over $A$. But $A$ is integrally closed in its field of fractions and hence $f \in A$. This proves that $j_\ast\mu_n = \mu_n$.