The canonical example of a divergent sequence that is square summable, $\sum_{n = 1}^\infty a_n$ is finite, is the harmonic sequence: $\sum_{n = 1}^\infty 1/n$.
Are there examples of sequences that also satisfy this property outside of the harmonic sequence that also satisfies these properties (divergent + square convergent)?
Yes, of course. See stuff like $\{b_n\}_{n\ge 10}$ where, to name a few examples \begin{align}b_n&=\frac1{\sqrt n\log n}\\ b_n&=\frac1{\sqrt{n\log n}\log\log n}\\ b_n&=\frac1{\sqrt{n\log n}(\log\log n)^{2/3}}\\ b_n&=\frac1{\sqrt{n\log n\log\log n}\log\log\log n}\end{align}