From the context, it appears that by "distribution function" the OP meant the pdf (usually, the distribution function is understood as the cdf). Let $F$ and $G$ denote the cdf's of $X$ and $Z_1$, respectively, so that $F=G^n$. (Note that $1-(1-G)^n$ is the cdf of the minimum, not the maximum, of $Z_1,\ldots,Z_n$. Thus, ofer zeitouni actually showed that the pdf of $\min\{Z_1,\ldots,Z_n\}$ is log concave.)
Let us show that the pdf $f=F'$ of $\max\{Z_1,\ldots,Z_n\}$ is log concave.
We have $f=F'=nG^{n-1}g$, where $g:=G'$, the standard normal pdf.
Next,
\begin{equation}
G^2(\ln f)''=-(n-1)gh-G^2,\quad\text{where } h(x):=xG(x)+g(x).
\end{equation}
Moreover, $h''=g>0$ and $h(-\infty+)=0=h'(-\infty+)$. So, $h>0$ and hence $(\ln f)''<0$, on the entire real line. Thus, the pdf $f$ is indeed log concave.
Remark. For any $j=1,\dots,n$, let $f_j$ denote the pdf of the $j$th order statistic $Z_{n:j}$ for $Z_1,\ldots,Z_n$, so that $Z_{n:1}=\min\{Z_1,\ldots,Z_n\}$ and $Z_{n:n}=\max\{Z_1,\ldots,Z_n\}$. It follows from a known formula for the pdf of an order statistic (see formula (2.1.6) on page 10 in David--Nagaraja) that \begin{equation} f_j=\binom{n-1}{j-1}f_1^{\frac{n-j}{n-1}}f_n^{\frac{j-1}{n-1}}. \end{equation} So, combining the log-concavity of $f_1$ proved by ofer zeitouni with the log-concavity of $f_n$ proved in this answer, we obtain the more general fact that the pdf $f_j$ of any order statistic $Z_{n:j}$ for $Z_1,\ldots,Z_n$ is log concave. Moreover, by shifting and rescaling, we now see that the pdf of any order statistic for any iid normal sample is log concave.