This can be done
with three copies. (There are 20 squares, and we need the total area to be divisible by 6 if we want a whole number of squares to fit on each face, so 3 is the minimum. If we don't have a whole number of squares on each face, there will be partial squares on faces, and the pieces won't fit together perfectly.)
To figure out how to do it,
the natural place to look is at the orange circle in this drawing:
That lower-right hole looks like it'll be hard to fill. But if we can put a vertex of the cube at the orange circle, that means we can bend the shape along the orange arrow, so it occupies the dark gray line. Once that's in place, a similar maneuver can be done at the pink circle to bend the upper right side down, compacting the entire shape into a single blob. And now, this can be done with three copies of the shape, making the yellow circled points of each shape touch at a vertex.
Here are some images of the covering:
Front view (focused on the "orange circle" fold):
Top and bottom view (focused on the "yellow circle" vertex, and the point directly opposite that)