National selection manager Kim Littlejohn said: "Daniel Flynn is having a very successful Plunket Shield with the bat for Northern and is likely to join Martin Guptill at the top of the order. He was part of the team in late 2009 and we're sure he'll make the most of his opportunity to again represent his country.
"We're mindful that Dean Brownlie is only just making a return to competitive cricket following injury and will keep a close eye on his progress. Although he's been included in the squad, his selection for the test will largely depend on how he pulls through the match for Canterbury this week.
"It's disappointing for Rob and Brent but at this stage we don't feel they've managed to cement their place in the squad. Both players will benefit from a return to domestic cricket where they can put performances on the board. We're sure they will do this and will be very much in contention for selection on the upcoming tour to the West Indies."
New Zealand need to win this week's test, which gets underway on Friday, to level the series after the first five-day encounter finished in a draw last week.
Their top-order batting has been ruthlessly exposed by the South African pace attack in the first two tests and the side will need to show better resolve at the Basin Reserve.
Right-armer Vernon Philander has been near-on unplayable for the visitors and has claimed 15 wickets in the first two tests, while Dale Steyn has picked up seven scalps and Morne Morkel has also been a handful.
Guptill, who has been in dominant form during the season in the limited overs side of the game, has struggled during the series and is yet to pass 22 in the first two tests and is averaging 11.25.
Wellington pace bowler Mark Gillespie, who was a shock inclusion for the second test but picked up a first-innings five-wicket bag, will be keen to back up his strong performance on his home ground.
The New Zealand squad will assemble in Wellington on Tuesday.
Team: Ross Taylor (captain), Trent Boult, Doug Bracewell, Dean Brownlie, Andrew Ellis, Daniel Flynn, Mark Gillespie, Martin Guptill, Chris Martin, Brendon McCullum, Tarun Nethula, Kruger van Wyk, Daniel Vettori, Kane Williamson
Follow Daniel Richardson on Twitter