According to ABC 7, the incident started when someone in the family placed a birthday cake in the overhead bin, which is not allowed.
Shortly afterward, the Port Authority Police Department was called in and everyone on the plane had to get off, according to the JetBlue statement. Despite reassuring the family that "no one did anything wrong", the airline forced the family to deplane and rebook their flight.
The family's luggage, however, made it to Vegas, leaving Mr Burke fuming.
The Burkes were then asked to leave the flight, and when they refused, police officers from JFK Airport boarded to investigate, which can be seen in a video recorded by the family.
Their seven-year-old son, Cameron Jr, and nine-year-old daughter, Camille, appear scared as the officers question the family. "When she told me I had been non-compliant, then I said 'ma'am, had you been drinking?' because her behavior was not normal", Cameron said.
"She then asked me to move it to underneath the seat in front of me", Cameron Burke told the newspaper. "Couldn't wait to get to Las Vegas, and all of a sudden this occurred, and my two children are screaming, crying", Minta said.
"You know, you could see the gestures - then she was pointing to her, did you tell him he couldn't put anything in the overhead compartment?", Cameron said. Things apparently escalated when the second flight attendant spoke with the first flight attendant that was dealing with the situation.
"The customers became agitated, cursed and yelled at the crew, and made false accusations about a crewmember's fitness to fly", the airline said in a statement (via People). "They're confused not knowing what's going on - they were traumatized", she said.
JetBlue claims the visual evidence that's emerged doesn't tell the full story.
The airline claims the family stored it in areas that were "improper" - an overhead bin for safety equipment - and refused to move it.
Cameron Burke's cellphone video shows his frightened family confronted by a Port Authority officer.
David Dao and United Airlines reached an "amicable" agreement after he was filmed being dragged off a flight.
Earlier this month, a Southern California father posted video showing him and his family being booted from a Delta flight after refusing to give up a seat.
That was when the flight attendants and the family allegedly began arguing over where the cake should be kept. Cameron Burke had reportedly purchased both the cake and the trip as a surprise to his wife Minta for her birthday, and the plan was to meet up with family at their destination.