Assume that $AB$ is on the line $y = 2x - 17$ and the coordinates of the other two vertices on the parabola are $C(x_1, y_1)$ and $D(x_2, y_2)$. Then $CD$ is on a line $L$ whose equation is $y = 2x + b$. Combining this with the equation of the parabola, we get $x^2 = 2x + b \Rightarrow x_1, x_2 = 1 \pm \sqrt{b+1}$. Assume that the length of one side of the square is $a$. Then
$$\begin{aligned} a^2 &= (x_1 - x_2)^2 + (y_1 - y_2)^2 \\ &= 5(x_1 - x_2)^2 = 20(b+1). \qquad (1) \end{aligned}$$

Pick a point $(6, -5)$ on the line $y = 2x - 17$, and the distance from the point to the line $y = 2x + b$ is $a$.
$$\text{So} \qquad a = \frac{|17+b|}{\sqrt{5}}. \qquad (2)$$
From (1) and (2), we get $b_1 = 3$, $b_2 = 63$, so $a^2 = 80$ or $a^2 = 1280$, and $a_{\min}^2 = 80$.