Welcome aboard!
The boolean operations result in closed paths.
You could convert the stroke of the spiral to a path before the intersection (Ctrl+Alt+C), although I have doubts you are after for that effect.
Instead, try grouping the spiral (Ctrl+G),
clone the square (Alt+D),
select both the group and the clone and set clipping.
This way you can alter the shape of the rendered area,
add more objects to be rendered inside the area and edit the spiral shape.