A St John Ambulance spokeswoman said staff treated a person in a moderate condition.
Passengers, including many young children just settled down to sleep, were taken off the plane when it landed in the early hours of the morning.
They were now checking back into a rescheduled flight due to leave at 5pm.
Those at the airport said it was chaotic with just two staff to tend to the plane-load of passengers.
"It's flipping bonkers here," said the passenger.
"Only two people on at Air NZ, one man screaming and swearing in line that he's going to sue, kids asleep everywhere."
Travel blogger Megan Singleton said her sister and brother-in-law were heading for a trip of a lifetime to Europe when their flight was suddenly diverted.
She said since landing, exhausted and frustrated passengers had endured lengthy queues through Customs and check-in as they prepared to leave the country for the second time in 17 hours.
Passengers were even forced to return duty-free goods, which was unheard of, she said.
The problems were compounded by passengers being told to rebook by phone, which was proving impossible because of swamped lines.
"There's chaos," said Singleton.
"It's ridiculous to be told to phone an 0800 number to rebook themselves."
Passengers had now been given vouchers to stay in hotels until their flight left later today.