The maximum of the function counting square roots is attained at $x_0=1$ and this statement generalises quite well.
Let $s(\chi)$ denote the Frobenius-Schur indicator of the irreducible character $\chi$. Thus, $s(\chi)=1$ if the representation of $\chi$ can be realised over $\mathbb{R}$, $s(\chi)=-1$ if $\chi$ is real-valued but the corresponding representation is not realisable over $\mathbb{R}$ and $s(\chi)=0$ if $\chi$ is not real-valued. Then, the number of square roots of an element $g$ in any group is equal to $$\sum_\chi s(\chi)\chi(g),$$ where the sum runs over all irreducible characters of the group.
Of course, in $S_n$, all Frobenius-Schur indicators are 1, so the number of square roots of $x_0$ is just $\sum_\chi \chi(x_0)$. This proves that the maximal number of solutions is indeed attained by $x_0 = 1$, since each character value attains its maximum there. This generalises immediately to any group, for which any representation is either realisable over $\mathbb{R}$ or has non-real character, e.g. all abelian groups. Other cases would require more thought, but this is likely the right way to go about it.
Edit: One reference I have found for the identity expressing the number of square roots in terms of Frobenius-Schur indicators is Eugene Wigner, American Journal of Mathematics Vol. 63, No. 1 (Jan., 1941), pp. 57-63, "On representations of certain finite groups". Once you get used to the notation, you will recognise it in displayed formula (11). Since the notation is really heavy going, I will supply a quick proof here:
Claim: If $n(g)$ is the number of square roots of an element $g$ of a finite group $G$, then we have $$n(g) = \sum_\chi s(\chi)\chi(g),$$ where the sum runs over all characters of $G$ and the Frobenius-Schur indicator of $\chi$ is defined as $s(\chi)=\frac{1}{|G|}\sum_{h\in G}\chi(h^2)$.
Proof: It is clear that $n(g)$ is a class function. So let's just take inner products with all characters of $G$ to find their coefficients, noting that we can write $n(g) = \sum_h \delta_{g,h^2}$ (here $\delta$ is the usual Kronecker delta): $$\left< n,\chi \right> = \frac{1}{|G|}\sum_{g\in G}n(g)\chi(g) = \frac{1}{|G|}\sum_{g\in G}\sum_{h\in G}\delta_{g,h^2}\chi(g) = \frac{1}{|G|}\sum_{h\in G}\sum_{g\in G}\delta_{g,h^2}\chi(g) = \frac{1}{|G|}\sum_{h\in G}\chi(h^2) $$ as required.