Born and raised in Hartford, Connecticut, Oszzo was the second youngest of eight children. Losing six of his siblings at young ages, he and his only surviving sibling, his younger sister Marcia, were very close. Oszzo attended Weaver High School and was known as a "ladies man" and an outstanding athlete. He was voted "most athletic" because he excelled on the basketball, football and track and field teams. He joined the United States Navy after graduating from high school and during his tour he continued playing basketball. He was honorably discharged as a first class machinist.

Shortly after his discharge, Oszzo married the love of his life, Joyce, whom he affectionately called "Lucy". They moved to Philadelphia, where they raised their family consisting of four loving daughters.