When a point $(x,y)$ is reflected across the $x$-axis, the image is $(x,-y)$. Thus the vertices of the square $(0,0)$, $(6,0)$, $(6,6)$, and $(0,6)$ are transformed to $(0,0)$, $(6,0)$, $(6,-6)$, and $(0,-6)$, respectively. The horizontal distance between $(0,0)$ and $(6,0)$ is $\boxed{6}$ units, so the sides of the reflected square are 6 units.
