Gutierrez was born in Havana, Cuba, the son of a pineappleplantation owner. As a successful businessman, his father was deemed an enemy of the state by Fidel Castro's regime. Faced with the expropriation of their property following the Cuban Revolution, Gutierrez's family fled for the United States in 1960 when he was six years old. Like many other Cuban American exiles, they settled in Miami. When it became apparent they would not be going home, Gutierrez's father accepted a position with the H. J. Heinz Company in Mexico and later started his own business. Gutierrez learned his first words of English from the bellhop at the hotel where they initially stayed and, some years later, he and his family acquired United States citizenship.