Lianne Killick, who was travelling with her husband Al and 9-year-old son Couper, described it as "really insulting".
"Some of the comments made by staff over there were almost like, 'what are you complaining about, you're on a tropical [island]'," she said. "We were in 30C heat, no air conditioning and queue after queue after queue in an airport. We weren't on a beach having pina coladas."
Another passenger said his wife, who had recently received a breast cancer diagnosis, had missed surgery because of the "debacle".
"We have not slept or rested properly for days now and remain extremely angry and upset," he said. "I am astounded that Air New Zealand seemed completely unprepared for an eventuality such as this. I still refuse to accept that a multi-million dollar national carrier has never encountered a mechanical failure requiring they support their fare-paying customers in a foreign environment."
Greg Britt, from Christchurch, said it was a "pretty hard" three days for all passengers.
"People think we had an extra few days off in Hawaii, but there was no time off," he said, describing everyone as "shot" with tiredness. "Kids were screaming for hours. Elderly people were getting faint, it was just horrific really."
Air NZ's chief executive has admitted the airline "failed more than 200 customers".
In an internal memo to staff, Christopher Luxon said the carrier had "let them down and more importantly let ourselves down".
"As chief executive officer I am ultimately accountable for this," he said, vowing to oversee personally a detailed review of the delay.
Meanwhile, an Air NZ flight from Hong Kong was delayed for nearly 20 hours after also suffering an engineering issue. Flight NZ80, with 284 passengers, was due to arrive in Auckland at 4.20 this morning. APNZ