Absolutely incorrect. Unfortunately many, including espn's Chris Mortensen, have been saying exactly that - that Mankins needs to play those six games in order to become a free agent. Sadly many others have said/repeated the same misinformation.
Mankins has five years of accrued service time. The rules for an uncapped 2010 were that a player needed six years to become an unrestricted free agent. The rules for an uncapped 2011 are five years to become an unrestricted free agent. Prior to that the previous CBAs (2009 and before) permitted a player to become an unrestricted free agent after four seasons.
So if there is no new CBA, Mankins is an unrestricted free agent. If on the other hand there is a new CBA, then the NFLPA would have to agree to lengthening the amount of time to becoming an unrestricted free agent by not one, but two years beyond what it had been in previous CBAs; I just don't see why they would agree to do that.
So the bottom line is that, no, Mankins does not have to play this year to become an unrestricted free agent.