Clifton leveled Hischier with his left shoulder with four minutes remaining in the first period of the Sabres' 5-4 loss to the Devils on Friday in Newark, New Jersey. Clifton received a match penalty following the hit on Hischier, who returned for the second period before sitting out the third.
Devils coach Lindy Ruff said Hischier will not play in the team's game against the visiting Minnesota Wild on Sunday due to an upper-body injury.
Clifton, 28, has one assist and a minus-6 rating in eight games this season.
In his first season with the Sabres after five with the Boston Bruins, Clifton has 44 points (10 goals, 34 assists) and a plus-31 rating in 240 career games.