I am not at all familiar with Rollo May's work (I just looked at the wikipedia entry to get grasp on who he was). In philosophy, the term "dialectic" can have several meanings. One meaning is just two things that engage each other -- think "dialogue." Another meaning is a dialectical method -- meaning that you arrive at a resolution by the interaction of two things.
Based on the general information about the author, I think that the meanings for Hegel, Kierkegaard, and Sartre will be pertinent here.
For Hegel, philosophy is "dialectical movement." For him, the entire process of human thought is our attempt to understand our world. In the process, we are dialectically related to the world. In other words, let's say we start with "Sense Certainty" -- which occurs near the beginning of his text Phenomenology of Spirit. Sense Certainty is the belief that what I see right in front of me is true. Dialectically, this turns out to be inadequate, because as you turn your gaze to different places, you see different things. This would mean that the other things are no longer true, but they are. Thus, definitions, on Hegel's picture, dialectical in nature (there is push and give in coming up with better ones).
Long story short, it turns out that what we are really facing for Hegel is a problem of self-definition where we are Spirit seeking to define the Absolute. There's a dialectical relationship between these two because we are conscious and then we realize self-conscious. And this means for us to be us, we need a dialectical other that interacts with us.
My advisor often said that the philosophers after Hegel are Hegelians without the whole.
Now to Kierkegaard, Kierkegaard through his pseudonyms is highly critical of the Denmark-version of Hegel's philosophy. For him, there is an existential dialectical between the self and God. In Anti-Climacus's work, this becomes clear as the self that relates itself to itself and in relating to itself relates itself to another. In other words, to be a self is to be dialectically related to the power that gives the self selfhood and existence (these are quotes from memory of the early part of Sickness unto Death which has the subtitle: "A Christian existential psychology").
Sartre agrees with Kierkegaard except insofar as he doesn't think there is a power outside the self that gives the self selfhood. Instead, we wholly define our meaning through our actions. At the same time, our definitions are not unconstrained. In Being and Nothingness, a major section treats the idea of the "Gaze" which is the experience of being a self for Others. (Cf. for Hegel, we are a self for the slave as master through the slave's identity-giving relation / for Kierkegaard, we are selves when we rest in God).
One consequence of this is that the Other for Sartre is a threat to the self's identity and freedom in a way that it is not for Hegel's all-encompassing social vision or for Kierkegaard's Augustinian self.
From my understanding, Christian existentialist psychology / philosophy was often called "personalism." But in both cases, the idea is that you need to find yourself in dialectical relationship to others.