  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 $\left(G_n\right)_{n \geq 0}$ given by the
recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ 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 $\mathcal{G}_{G_0, G_1}^2\!(k)$, 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.
