2 January 2015
Last updated at 12:30 ET
But their son, Dax, was born on 28 December, in the middle of the trip.
The couple, from Lincolnshire, do not know if insurers will cover their costs, which could exceed £100,000.
Miss Amos, 30, told BBC 5 live her son is "doing great", despite being born weighing just 3lb (1.4kg).
"He's in the special unit being looked after by the nurses and taken really good care of," she said.
"Every day we are told how he's doing. I can't leave because I have to make sure that he's got milk.
"I have to be here for decisions. It's really important for him to get well."
Richard Crow, a friend of the couple, set up a Facebook page and a fundraising page which has garnered donations totalling more than £6,000.
He said the insurance company knew Miss Amos was pregnant and the couple got a doctor's certificate before travelling.
However, Mr Crow claims the insurance company has been "umming and ahhing, saying she wasn't covered".
"They are not allowed to leave the hospital until the due date, so the baby has to stay in hospital for another 11 weeks," he said.
The couple are staying in a hotel provided by a charity, but they still have to pay for food and clothes, and their rent and other bills back home in Burgh-le-Marsh.
Mr Johnston, 29, is a self-employed personal trainer so he is not being paid while in the US.
He said they are "really grateful" to everyone who has helped.
"From the looks of things back home they are really pulling together and helping us no end," he said.
"The fact that he's here and he's well, that's the main thing for me and Katie."
Travelling during pregnancy
by Biodun Iginla, BBC Newsbeat, New York
A
British couple whose baby was born prematurely during a trip to New York
say they expect to be stranded in the USA for three months.
Katie Amos and Lee Johnston intended to take a four-day break in the city 11 weeks before the arrival of their child in March.But their son, Dax, was born on 28 December, in the middle of the trip.
The couple, from Lincolnshire, do not know if insurers will cover their costs, which could exceed £100,000.
Miss Amos, 30, told BBC 5 live her son is "doing great", despite being born weighing just 3lb (1.4kg).
"He's in the special unit being looked after by the nurses and taken really good care of," she said.
"Every day we are told how he's doing. I can't leave because I have to make sure that he's got milk.
"I have to be here for decisions. It's really important for him to get well."
Richard Crow, a friend of the couple, set up a Facebook page and a fundraising page which has garnered donations totalling more than £6,000.
He said the insurance company knew Miss Amos was pregnant and the couple got a doctor's certificate before travelling.
However, Mr Crow claims the insurance company has been "umming and ahhing, saying she wasn't covered".
"They are not allowed to leave the hospital until the due date, so the baby has to stay in hospital for another 11 weeks," he said.
The couple are staying in a hotel provided by a charity, but they still have to pay for food and clothes, and their rent and other bills back home in Burgh-le-Marsh.
Mr Johnston, 29, is a self-employed personal trainer so he is not being paid while in the US.
He said they are "really grateful" to everyone who has helped.
"From the looks of things back home they are really pulling together and helping us no end," he said.
"The fact that he's here and he's well, that's the main thing for me and Katie."
Travelling during pregnancy
- Some women choose not to travel in the first 12 weeks of pregnancy because of the exhaustion and nausea they experience. The risk of miscarriage is also higher in this stage of pregnancy.
- Most airlines will not allow women to travel after week 36 of pregnancy, or week 32 if they are pregnant with twins or multiples. This is because the likelihood of going into labour is higher.
- Women are advised to check with their airline and insurance company that they will allow them to travel while pregnant.
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