Strongest LEGO structuce - (1) Pillars
I am also going to recommend Technic as the stronger support option, but I would make the case out of "normal" Lego, and then implement 2 Technic rectangular "rings" around the case at the place of your pillars, joined at the top and bottom (on the top the joint could form the handle). Make these rings or braces by overlapping Technic pieces and joining them with Technic pins. Then, in your case, insert a few pieces that can also accommodate a Technic pin (1x2 brick with center hole) and secure the brick-built case to the Technic rings that way. If you think this through, you could make that connection easily removable, so that you can slide the brick-built case out of the 2 rings and have an opening side panel allowing access to the internals - a very flexible but still very strong design. 

I have mocked up this case, see the below pictures. No comments on the sloppy building style - I had to work with whatever my son did not have incorporated into one of his builds :). The Technic rings obviously need some more bracing in your final design, but I think this shows the concept of using 2 rings, a handle, and pins to secure and remove the central case pretty well: