The point in using a softbox is to create a softer light, that means, a light that creates soft transitions from lighted areas to shadows. By the laws of physics and how light works, what defines whether a lightsource is soft or hard is its relative size to the subject being illuminated.
In a softbox the diffuser (the white fabric in the front) is meant to create a large area that irradiates light. The reflective material inside the lightbox is meant to redirect light that would otherwise spread in all directions towards the diffuser, optimizing the power of the lightsource and avoiding light leakage (ligth falling where you dont want/need it).
When the light comes from a large area, there will be parts of the subject lit only by a fraction of the rays coming from the source. Say, part of the face may be sadowed from one corner of the lightbox, but not the other three. There will be other areas shadowed from half the lightbox, and so on, until you reach an area of the subject that is completely shadowed from the box.
For a couner expample, think for a moment on the light from a single candle. In a room illuminated by no other source than a single candle, the shadow you would project into a wall, would be very defined, almost pencil-sharp. So would be she light-shadow transition in a face, illuminated from a single candle. Put it for example axactly to one side of the person. So one cheek is illuminated and the other is in the shadow. Amost for sure, the nose of that person would be illuminated on one half and completely in the shadow in the other half, with a very sudden transition (almost no area of the nose would be half-lit).
You see that the flame of a normal candle is very small, compared to a human face. It works as a hard light because almost all the rays come from a very little area, amost a single point in space. (it is a puntual source).
Now think of the largest area possible as a light source we can have access to: a fully cloudy sky. Light comes from everywhere, so you can't even project a sadow, because even if you cover the rays comming from one side, rays comming from somewhere else lessen the effect.
In the case of the equipment you are proposing to buy, the working set of tubes and reflector surfaces create a large area that "irradiates" light, it is apparently much larger than a face (it seems to be 24 x 24 inches). There are lightboxes of that size, wich means that you would'n gain anything adding it to this gear. Of course there are larger softboxes, but they aren't made for this type of lamp. And the larger soft boxes get pretty expensive, to the point that you may be better buying a second unit and placing them side by side (doubling the radiating area, thus creating a softer light).
If you really wan't to stick with this unit, you may get good results using it as it is, but if you feel you need a diffuser, you may do a special diffuser for your unit by yourself. There is a type of fabric called "taffeta". It is used for making chlothes, but the white one is very similar to what some light modifiers use. You may make a sort of "shower cap" but for the size of this light head. It would be really easy to install and remove, ans you would be able to fold and store it in the same cage/carrying case. Another option would be to stick or sew velcro in the border of the barn doors and in the dorders of your taffeta diffuser.
There are other materials you could use instead of taffeta, for example white vinyl shower curtains, or several other fabrics used for bed sheets and table dressing. (Just cut and sew appropriately and it will look pro).