"I think he just wanted to be a New Yorker"

It was supposed to be a quick holiday visit to New York for British couple Lee Johnston and Katie Amos, but last week, it turned into a medical adventure when Amos gave birth to their first child as she was just beginning her third trimester.

Amos was unaccustomed to American hospitals and health care. She worried that she and her fiance would be stuck with bills they could not afford.

While they do have to manage the expense of living in New York for the next few months, one thing they don't have to worry about are the hospital bills, reports CBS News correspondent Michelle Miller.

Their New York adventure started two days into their trip. Amos suddenly went into labor - 11 weeks early - while the couple was on a tour of Manhattan's Central Park. They headed to .

It was in the taxi heading to Lenox Hill Hospital on New York's Upper East Side that Amos and Johnston realized they might be in a predicament.

"I looked at you and I went, 'Maybe these are contractions,' and you just looked at me and went, 'Really?'" Amos said, laughing. "When we went into the hospital and got checked out and I was laid on that bed and they were checking me and they were like, 'Yeah, you're dilated,' and I was just like, 'Ah. I'm having a baby in New York.'"

Amos delivered baby Dax, all 3 pounds, 4 ounces of him, and the hospital staff rushed him to the neonatal intensive care unit.

"I just said, 'I want to go home,' and they said, 'You're not going home,'" Amos said.

At nearly three months early, Dax still has to develop under the supervision of doctors. That can take time, according to Amy Marshall, the nurse manager for the NICU at Lenox Hill.

"So in order to go home, he needs to obviously grow a lot bigger, get off any respiratory support, and then learn to eat on his own," Marshall said.

That means the family could be in New York until March, closer to the baby's actual due date. Amos and Johnston are both self-employed personal trainers, now with no money coming in. Lisa Schavrien, a nurse navigator at Lenox Hill, has helped to guide the couple giving birth in New York.

"This is quite extreme, I mean she is 29 weeks pregnant and lives out of the country," Schavrien said.

A friend back home created a Facebook page in Dax's honor and began a crowd-funding campaign to help the couple cover the costs of their stay. This weekend, it was settled that the hospital accepts their health coverage.

"We were always going to accept what their insurance could pay, and then other than that, there's not going to be any out of pocket for them because we need to accept what the insurance would cover for them," Schavrien said.

Through their British health insurance and travel insurance they purchased, it's now not their medical bills giving the couple pause. Rather, it's the expense of living in New York.

Lenox Hill helped the couple secure housing near the hospital, along with some other basics.

"I myself went to my apartment and got a whole bunch of clothes for her, to take it over to her, because I knew she didn't have anything, and we're relatively about the same size. Unfortunately he doesn't wear my size so I really couldn't get him any clothing," Schavrien said.

Though the couple only packed for a short trip, they've already been through quite an adventure.

"He needs to be well enough to fly back," Johnston said.

"So possibly keeping him in up to his due date," Amos said.

"Which is the 10th of March, so we've got a way," Johnston added.

The couple said they'll be right by Dax's side.

There's one more bonus about Dax's unusual birth.

"He's got dual citizenship, hasn't he?" Johnston said.

Amos called situation "crazy."

"Honestly, I think he just wanted to be a New Yorker," she said.

Baby Dax is showing very good signs of health.