The 25-year-old picked up the problem in last month's 2-1 defeat at MK Dons after a challenge with Jemal Johnson and has missed the last five matches.
Colchester physio Tony Flynn expects Beevers to be out for "months rather than weeks" and he can only begin rehabilitation when the effects of the surgery wear off.
Despite the setback, the former Lincoln man says he is focusing on his recovery and forcing his way back into manager Aidy Bothroyd's side following a solid start to life at the Community Stadium.
He told the Daily Gazette: "I've settled in really well at Colchester and it's unfortunate with the timing of it, with the new manager coming in.
"He said he was disappointed for me but that he wants to see me back as soon as possible."