I appreciate wanting to MacGuyver a modifier but in this case, it shouldn't break the budget to buy either a) an umbrella or b) a proper soft box. I use a 16 x 20 Chimera and with the proper speedring and stand attachment, does the majority of my run and gun shooting. A small ballhead allows positioning.
As well, cages won't pack down at all which would make the exercise kind of pointless imo. Would suck to put a tomato cage in the trunk only to later find it has punctured your expensive GoreTex jackets or worse, damage your gear.
Basically, embrace the lights and modifiers cuz one day it's gonna save your bacon.
If you're committed to trying to make a box, as noted elsewhere, foam core would be the easiest route. Attaching this to a light stand and flash is another hurdle as I wouldn't risk attaching the hack box to a $500 speedlight.
Good luck and post some photos if you do end up making a soft box