"Writhing in pain inside a dilapidated hospital in the Dominican Republic, Brooklyn native Jessica Mazzone didn't think she'd ever walk again.

The 28-year-old flight attendant had fallen three stories from a hotel balcony on the eastern tip of the island, snapping her spine and shattering her left leg.

Local doctors insisted they had to operate on her at the hospital. A journey to the U.S. would almost certainly leave her paralyzed, Mazzone was told. Her family feared the worst...