jmc

First, we divide both sides of the equation by $2$ to find that $x^2+y^2-6x+2y=10$. Completing the square, we arrive at $(x-3{)}^2+(y+1{)}^2=20$. Thus, the circle has radius $\sqrt{20}$.



It follows that a side length of the square is equal to the diameter of the circle, or $2\sqrt{20}$. The area of the square is thus ${\left(2\sqrt{20}\right)}^2=\boxed{80}$ square units.

Notice that the information that the sides were parallel to the x-axis is irrelevant.