Born and raised in St. Louis, Missouri, Bukich was the son of Croatian immigrants. He was starting quarterback in his senior year at Roosevelt High School, a team that did not win a game. Bukich played one game as wingback when Roosevelt tried to upset eventual league champions [[Cleveland High School, St Louis] by attempting to surprise them by playing the single wing.
Bukich earned a football scholarship to the [[Iowa State Cyclones football|Iowa State University] in Ames, Iowa as a wingback, and transferred to the University of Southern California in Los Angeles. During his senior year in 1952, USC played in the Rose Bowl on New Year's Day. After the Trojans' primary passer, All-American halfback Jim Sears, was sidelined early in the first quarter with a broken leg, Bukich came in and conducted a drive, completing all but two of his passes, that resulted in the only score of the game, and USC defeated Wisconsin, 7–0. Bukich was selected as the Most Valuable Player in the game, and after his career, he was inducted into the USC Trojan Hall of Fame and the Rose Bowl Hall of Fame (2004).
Bukich's greatest performances were during his second stint with the Bears in the 1960s. He was a reserve quarterback with the Bears in 1963 when they won the NFL championship (Bill Wade was the starter throughout the season). Bukich replaced Wade during Week Four against the Baltimore Colts after Wade completed just 5 of 21 passes. Bukich played the remainder of the game, completing 6 of 7 passes with a touchdown pass to Ronnie Bull, which was the deciding margin of victory.