I haven't seen the movie, but I have heard that Nixon was an ENTJ in real life. I have read assessments of his personality and he appeared to be an E TJ that was very low in agreeableness. I didn't have enough information about the S/N.
I thought about ENTJ too.
However, his concerns seemed to be more like that of a Keirsey Guardian than a Keirsey Rational.
However, at the end of the movie, he tells Frost that he wasn't suited to be a politician because it turned out to be more than the issues, getting things done, and intellectual debate. I could see that being an ENTJ concern..