The girl band was due to play a matinee and evening show at the SSE Arena on Thursday.
The cancellation was announced just minutes after the doors opened for the afternoon performance.
Fans were later informed by text message that the show would not be going ahead as scheduled.
Michael Green was in Belfast for the concert with his daughters, aged five and six.
He said the family had been among fans who were disappointed when One Direction re-scheduled their Belfast concert last October.
Mr Green told the BBC he felt "physically gutted and a little bit sick" when he was told the kids were to be let down again.
"I thought this couldn't happen again, what are the chances?"
Michael had surprised his daughters with the tickets on Christmas morning and travelled from Coleraine to see the girl band.
He said he was worried the band would re-schedule for an afternoon when the children were back at school.
"I'm really, really annoyed. Once again they've waited until we got here to tell us it was cancelled."
Mother Geraldine McLaughlin had brought her stepdaughter, Stacey, to the concert for her birthday.
She said the 16-year-old's birthday had been ruined by the news.
"We came down to Belfast and booked into the Ramada to make a day of it," she said.
"Stacey was so excited about this, I don't know what we'll do for the rest of the day.
"We're just going to go back to the hotel now to get away from this chaos."
Thousands of young people were gathered at the SSE Arena at lunchtime on Thursday when the announcement was made via speakerphone.
Ms McLaughlin said many of them were in tears.
"It's not been handled very well," she said. "There's absolute bedlam down here now.
"Surely they could have informed us before we all landed at the place? This was minutes before the show was due to begin."
A statement from gig promoter Aikens said: "We regret to announce that both Little Mix shows at the SSE Arena will not take place today due to Jesy being unwell and not able to perform.
"Please hold on to your tickets - we hope to be able to make a further announcement on possible rescheduling later today."
Ms McLaughlin pointed out that if the concert is rescheduled for next week, children will be back at school.
"You can't just take a day off school and time off work for a concert," she said.
Mother Sarah-Louise Murray had tickets for her and her three-year-old daughter, Scarlett, to see the band on Thursday afternoon.
She said Scarlett had been in tears when she was told the concert was not going ahead.
"We had special matching T-shirts made for the day and she was so looking forward to it.
"We arrived after 13:00 when the doors were due to open because I wanted to avoid the crowds, but when we got to the arena the queues of people were still waiting outside.
"When we heard from others that it was cancelled I put Scarlett back in the pram and told her we had to go home and come back another day."
It is the second major cancellation to disappoint young music fans in the city in recent times.
Last October, One Direction called off a gig at the SSE at late notice after singer Liam Payne became ill.
The band Little Mix have been forced to cancel their Belfast gigs after singer Jesy Nelson became unwell.