Do the Big Five traits interact?
I have given a general overview of the Big Five model in a previous answer. The Big Five model of personality is based on a statistical analysis of a person's tendency to agree or disagree with short descriptive statements. The principal object of this analysis is to identify independent factors that can explain variation in participants' responses to these statements. In some sense, they are designed not to interact.
Because of the descriptive nature of, and the methodology employed in, the construction of the Big Five, the factors do not have notion of interaction in the sense that Jungian types might, and do not "drive" behavior in the same sense. It's theoretically meaningful to talk about interactions between Jungian types, because types are associated with not only particular behavioral patterns, but particular manifestations of behavior.
Factor analytic theories such as the Big Five or 16PF, on the other hand, are not associated with particular manifestations of behavior. It isn't meaningful to talk about how a person A who is high on neuroticism might interact with a person B who is high on Agreeableness, because we can readily imagine an almost infinite number of ways in which those traits might manifest over persons, situations and time.
There may still be statistical interaction between Big Five traits with respect to some outcome variable (e.g., high conscientiousness and extraversion may predict higher income than the sum of both in isolation), but these notions of interaction should not be confused.
Are interactions between Jungian types scientifically supported?
This question has effectively been covered a number of posts here on Cognitive Sciences. The short version is that Jungian type theories such as the MBTI are not well supported by the empirical research literature, but we don't know if it is an example of the Forer effect. You may find the following questions interesting.