Coach Gary Pinkel did not have a diagnosis or definitive news on Beckner's injury but said, "We think it's an MCL. ... It could be more."
Pinkel also noted the team's medical staff would further evaluate the knee to determine the extent of the injury.
Beckner was hurt on BYU's first offensive possession of the game. Missouri trainers tended to him on the sideline before wrapping his right knee and having him carted off to the locker room.
The 6-foot-4, 300-pound freshman entered Saturday with 27 tackles on the season, including eight for loss and three sacks.
Beckner, a former five-star recruit, was ranked by ESPN.com's RecruitingNation as the No. 1 defensive tackle prospect in the 2015 class.