Electronic & Hydraulic Shifting (Di2, Ui2, customisations)

Depends what you mean by 'box'. You'll need a Junction A box at the front of the bike, either a 3 or 5 port SM-EW90 or an old school SM-EW67, but the older EW67 won't work with an internal battery. Depending on how your cables are laid out, you'll probably also want a Junction B at some point around the BB to connect cables together. So, assuming non-bonkers cable runs you'll need:

SW-R600 climbers switch
microswitches
RD6870 rear mech
Battery - either external or internal to suit
Junction A box - SM-EW90A or B if internal battery, any if external
Junction B box - to link cables together at the BB

You'll also need a charger for the battery. Obvs.

As for the cost, I'll let you Google that for yourself. I've got Cyclocross framesets to gaze at.