Can you double a square with a straightedge and a compass?

I'd say yes.  If you use the diagonal as a reference.  Take the square and set your compass to the width of the diameter of the square.  Now put it on the page and mark a point.  Put the point of the compass on that mark and make another mark.  Now you can connect the two marks with the straight edge and you have a line that, if you made a square with sides that long, it'd have 2x the area of the first one.  That's because the diagonal is the square root of 2 larger than one side.  Square the square root of 2 and you've got 2.  You lust need to make a perpendicular line to the first one to get the box going.