Regeczi enrolled at the University of Michigan where he won letters in football, basketball and baseball. Regeczi played in Harry Kipke's offense that won back-to-back national championships in 1932 and 1933. Regeczi had one of his best games in a 40-0 win over Cornell in 1933, a game in which Regeczi rushed for 101 yards. Regeczi gained the majority of his yards against Cornell on an 85-yard touchdown run from punt formation in the first quarter. Regeczi also played an important role in the 1933 Michigan-Ohio State game. The game was played at Michigan Stadium in front of 93,508 fans—the largest crowd to see a Big Ten football game to that date. Regeczi kept Ohio State at bay with his punting and also playing at the fullback position. While he was regarded as a good running back, Regeczi won his greatest acclaim as a punter. In October 1933, George Kirksey, writing for the United Press, said: "Regeczi, the Hungarian halfback, is probably the greatest punter for distance and accuracy in the country..." Regeczi suffered a knee injury at the start of the 1934 season and had difficulty regaining his speed after the injury. He was kept in the lineup in 1934 mostly due to his punting ability. The Associated Press reported in November 1934 that after three years in the starting lineup, Regeczi "found himself in the position of battling for a starting position." Due in part to injuries to Regeczi and quarterback Bill Renner, Kipke's offense in 1934 was sometimes referred to as "Punt, Pass and Prayer." A September 1934 newspaper story described Regeczi's role in the system:
"A punt, a pass and a prayer. By Harry Kipke's own words, that's all they have over there. But when it's a Regeczi punting, a Renner passing, and a typical Michigan squad of good ends, good tackles, good guards, and a great center 'preying', how are you going to stop them?"
Regeczi was teammates with Gerald Ford, who later served as the U.S. President, on the Michigan teams from 1932 to 1934. In 1994 interview, Ford praised Regeczi's punting in discussing the "Punt, Pass and Prayer" offense:
"In '32 and '33, we were undefeated, and then in '34 we had a tough, tough year. In those years, our offense was called a punt, a pass, and a prayer. We had an outstanding passer, Bill Renner, who broke an ankle before the season started. Our punter, John Regeczi, was the greatest college punter I ever saw and he ruined his knee. All we had left was the prayer."
Ford was selected as the Most Valuable Player on the 1934 team. Despite the injury and limited playing time in 1934, Regeczi was selected to play on the College All‑Star team after his senior year. Regeczi and Gerald Ford were the only two Wolverines selected to play on the 1935 College All-Star team. Although the Chicago Bears beat the College All-Stars 5 to 0, Regeczi's punting was singled out for praise in the press coverage of the game: "And the punt. The game showed this weapon the important thing in football. You noticed the Bears never came close to while Regeczi was kicking."
