Opener Guptill suffered a hamstring strain and had to retire hurt early in his innings in the first match in Hamilton, but that did not stop him returning to hit the winning runs.
McClenaghan will play no further part in the series after suffering a side strain which prevented him finishing his final over after he had taken four wickets at Seddon Park.
Test seamer Southee and Rutherford, who made his international debut in New Zealand's Twenty20 series defeat against England this month, come into the 13-man squad in time for Wednesday's second match in Napier.
McClenaghan was set to have a scan on his injury.
New Zealand physiotherapist Paul Close said: "Unfortunately Mitchell will miss the rest of the one day series, while Martin will be out for Wednesday night and we will assess him again today."
Southee has just regained fitness after ligament damage to his left thumb.