My guess from the way things have been depicted so far is that they will encounter each other while investigating the boxes (which I'm guessing could be micro boob tubes or tesseracts of somekind) while ruffling each others feathers along the way while learning to work together to solve the crisis at hand. Then after they are together they decide they can work together and form the Justice League. I'm guessing they nominate Superman as their leader and Bruce will give his "Gotham gets my priority but alert me if there is something actually seriously you need my help with and maybe I'll aide you guys" speech. Then the second arc flash forwards to the present and we see how the roster has expanded, etc.
The boxes are Mother Boxes ( as seen in the New Gods
) and they do open boomtubes. The ones seen look more like the version used by the Forever People than the cell phone size ones used by others from New Genesis.