Not quite. Kenny O'Donnell did exist, but his role in the crisis was beefed up for artistic reasons.
Love this flick, especially how they showed what an asshole LeMay was. As much credit as he's given for setting up SAC, everything I've read about him is unanimous in the opinion that he was an unmitigated ass. Favorite moment in the flick is when he gets into a heated discussion with Greenwood as JFK, and in pressing his point starts to walk up behind the president, but O'Donnell cuts him off. Greenwood turns around, and his eyes just flicker at O'Donnell, seeing that he moved into LeMay's path, and seemingly understands everything that just happened behind him. Just an eye flicker, no other response. Conversation continues without pause, as if nothing happened. So understated, but awesome.
EDIT: Oh, and while Greenwood was a stretch as JFK, huge, huge props to Stephen Culp as RFK. Just Erie how spot-on he was. Was the second time for him playing Bobby, and I don't think anyone will ever even come close to his performance. His "I'm not so ruthless" is just supreme.