GCD of sums of $k$ consecutive squares of generalized Fibonacci numbers

In 2021, Guyer and Mbirika gave two equivalent formulas that computed the greatest common divisor (GCD) of all sums of $k$ consecutive terms in the generalized Fibonacci sequence ${\left(G_n\right)}_{n\ge 0}$ given by the recurrence $G_n=G_{n-1}+G_{n-2}$ for all $n\ge 2$ with integral initial conditions $G_0$ and $G_1$. In this current paper, we extend their results to the GCD of all sums of $k$ consecutive squares of these numbers. Denoting these GCD values by the symbol ${\mathcal{G}}_{G_0,G_1}^2\text{\hspace{0.17em}⁣}(k)$, we prove $$\mathcal{G}${G_0, G_1}^2!(k) = \gcd\left(G_k G{k+1} - G_0 G_1,; G_{k+1}^2 - G_1^2,; G_{k+2}^2 - G_2^2\right)$. Moreover, we provide very tantalizing closed forms in the specific settings of the Fibonacci, Lucas, and generalized Fibonacci numbers. We close with a number of open questions for further research.