Applying Superposition Property : $x^{2}[n]$ vs $x[n^{2}]$
Given the system I/O definition:
$$y[n] = \mathcal{H}\{x[n]\} = x[n^2] \tag{1} $$
you can easily show that it's a linear (but time-varying) system.
Following the standard procedure
let
$$y_1[n] = \mathcal{H}\{x_1[n]\} = x_1[n^2] \tag{2.1}$$ and
$$y_2[n] = \mathcal{H}\{x_2[n]\} = x_2[n^2] \tag{2.2}$$
then define
$$x_3[n] = a x_1[n] + b x_2[n] \tag{3}$$
and
$$
\begin{align}
y_3[n] &amp;= \mathcal{H}\{x_3[n]\} \tag{4}\\\\
&amp;= x_3[n^2] \tag{5}\\\\
&amp;= a x_1[n^2] + b x_2[n^2]\tag{6}\\\\
&amp;= a y_1[n] + b y_2[n]\tag{7}\\
\end{align}
$$
where Eqs(4)-(5) follow Eq.(1), Eq.(6) follows Eq.(3), and Eq.(7) follows Eq.(2).
Equation (7) indicates that the system is linear.