The 24-year-old was poised to make his debut in Christchurch next week in the first of three Tests as a stand-in for Jamie Smith, who is on paternity leave.
But he suffered the injury ahead of play on day two of a tour match in Queenstown against a New Zealand Prime Minister's XI.
"I'm gutted for Jordan," England coach Brendon McCullum said.
"He has been shaping up nicely, both with the bat and gloves, since we got to New Zealand.
"That's sport, and unfortunately, these things happen. We will get around him and look after him. He is resilient, and his time will come at some point in the future."
Ollie Pope stepped in as wicketkeeper during the warm-up fixture with England yet to name a replacement for Cox.
The Test series starts in Christchurch on November 28th before moving to Wellington and then Hamilton.