Connor joins the Order in 1770 and we know that a major chunk of the game takes place in Boston. This is the year where anti-tax tensions boil over into civil unrest on the streets of Boston and New York. The most infamous example became the Boston Massacre where angry mobs attacked customs officials and began taunting and throwing rocks at a group of Redcoats. The British troops panicked and fired their weapons into the crowd. Five demonstrators were killed.
This quickly became a major propaganda victory for the Patriot cause and I theorize that this might be the turning point where Connor links up with the American Assassin's Order and begins fighting the British occupation. Hell, AC2 started off with the Pazzi Conspiracy so it wouldn't surprise me that another major historical event sets the events of AC3 off.