Perkins, the Celtics' starting center, missed his second straight game Monday night with a sore left shoulder. Allen, who comes off the bench, was sidelined for the fifth consecutive game with a strained right ankle.
O'Neal missed his eighth straight game with a sore right knee. Calderon missed the game after returning from a four-game absence Sunday, also against the Celtics, because of a strained right hamstring. Boston won that game 94-88 in Toronto.
Paul Pierce started for Boston despite hurting his knee in a first-quarter collision with teammate Brian Scalabrine on Sunday. He missed the final 7:25 of the first quarter then started the second and finished with 11 points as Boston broke a four-game losing streak.
"They're out at least until the early part of next week, maybe longer," Rivers said.
Scalabrine started in place of Perkins on Monday.
Rivers said he believed Perkins' shoulder was at least partially dislocated and could be a problem for the rest of the 24-year-old's career.
The shoulder "is going to go back out, you know that," Rivers said. "When and what day, we have no idea. But we know it will. It's just bum luck."
Perkins said he was feeling better.
"They want to give me more time to rest," he said before Monday's game. "I guess we're thinking long term. We know we're going to make the playoffs. I guess they'd rather have me for the playoffs."
O'Neal, who went on the inactive list Dec. 31, is averaging 14 points and 7.3 rebounds.
Calderon is averaging 12.9 points and has made 79 straight free throws dating to last season, the third-longest streak in NBA history. Minnesota's Micheal Williams holds the record of 97 in 1993.