Clifton, a native of Long Branch, New Jersey, was selected by the Phoenix Coyotes in the fifth round (No. 133) of the 2013 NHL Draft after playing with the United States National Team Development Program and helping the U.S. finish second at the 2013 IIHF World Under-18 Championship.

The defenseman played the next four seasons at Quinnipiac University (2013-17), where he was named captain as a junior and helped the Bobcats reach the 2016 NCAA Frozen Four before losing to North Dakota in the championship game. Clifton was named to the All-Tournament team that season.

After failing to reach an agreement with the Coyotes following his senior season at Quinnipiac, Clifton signed with Providence of the American Hockey League, the minor league affiliate of the Boston Bruins.

Clifton played 54 games for Providence in 2017-18, then was signed to a two-year contract by the Bruins on May 3, 2018. He began 2018-19 back in Providence but was called up six weeks into the season and had one assist in 19 regular-season games. Clifton's first NHL goal proved to be the game-winner in Game 2 of the Eastern Conference Final against the Carolina Hurricanes on May 12, 2019, a 6-2 victory.

Clifton signed a three-year, $9.99 million contract with the Buffalo Sabres on July 1, 2023, after setting NHL career-highs in games played (78), goals (five), assists (18), points (23), and plus/minus (plus-20) in 2022-23. He was traded to the Pittsburgh Penguins on June 28, 2025.