After the middle east, a lot of "democracies" are considering shutting down citizen's voices. India recently did so. Democracy's representatives find it inconvenient that citizens have a voice after elections.
Disconnecting the county's Internet from the wider Internet is not the same as 'turning off the Internet.' Citizens can still communicate with each other to organize, which is what you seem to imply that the New Zealand government is attempting to prevent.
The law would create a three strikes system for copyright infringement. Triple violators would be disconnected from the Internet without due process. It's not a national disconnect system, but could lay the technical grounds for one if implemented.
It wasn't done in the same bills related to Christchurch. Rather, during the 'overtime' to deal with the aftermath of the Christchurch quake, this bill was promoted quickly also.
Doing this speeds up the process and avoids a huge amount of oversight, consideration and public input, and has very much caught the NZ public by surprise.
It's unlikely we (NZ public) will get organised enough to change this, and past laws with significant public outcry (which this doesn't have yet) have stayed without review, as referendum's are not binding in NZ.