The second son of three boys, Greco was born on September 30, 1993, in Queens, New York[1] to parents Paul and Mary Jane.

[2] His mother is a retired nurse[2] while his father served as a member of the New York City Fire Department and suffered illnesses and conditions following his efforts during the September 11 attacks.

[4] He then turned pro, playing in the American Hockey League for the Bridgeport Sound Tigers[5] and Springfield Thunderbirds.

Greco was signed by the Florida Panthers of the National Hockey League to a two-year, two-way contact in November 2017.

On October 9, 2020, having left the Sharks as a free agent he joined the New York Rangers on a two-year, two-way contract.