Analog of the Birkhoff's ergodic theorem for the sequence of squares

No - the sequence of squares is universally bad which was proved by Buczolich and Mauldin. I will quote from Tom Ward's review of their paper Divergent square averages, Ann. of Math. (2) 171 (2010), no. 3, 1479–1530.

A consequence of J. Bourgain's work [Inst. Hautes Études Sci. Publ. Math. No. 69 (1989), 5–45; MR1019960] is an ergodic theorem along squares, answering earlier questions of Bellow and Furstenberg: If $(X,\mathcal B,T,\mu)$ is a measure-preserving system, then the non-conventional ergodic averages $$ \frac1{N} \sum_{n=0}^{N-1} f(T^{n^2} x) $$ converge almost everywhere for $f\in L^p$ with $p>1$. Here a comprehensive - and negative - answer is given to his question of whether the result extends to all of $L^1$. The authors show that the sequence $(n^2)$ is universally bad: for any ergodic measure-preserving system there is a function $f\in L^1$ for which the above averages fail to converge as $N\to\infty$ for $x$ in a set of positive measure.

PS The Birkhoff theorem does not apply to your ``particular case'' as it requires the presence of a finite invariant measure.

If $X=\mathbb{Z}$, $\mu$ is the counting measure, and $T$ is the shift operator given by $Tf(x)=f(x+1)$, then for all real $p\ge1$, $f\in\ell^p(\mathbb{Z})$, and $x\in\mathbb{Z}$, by Hölder's inequality, $$ |\mathcal{A}_N f(x)|\le \frac1N\,\sum_{n=0}^N|f(x+n^2)| \le\frac1N\,\|f\|_p\,(N+1)^{1-1/p}\to0 $$ and hence $\mathcal{A}_N f(x)\to0$ as $N\to\infty$.