If you haven't experienced the feeling that John Green is talking about, you have a treat in store for you. The time will come when a book will make such an impression on you that you will want everybody to read it. The key words in the quote seem to be "weird evangelical zeal." They simply mean that a book will sometimes make such a strong impression on you that you want to share it with everybody. Unfortunately, you are often disappointed by the reaction you get from others. It has to be the right book for the right person at the right time.
John Green is probably exaggerating for effect. It would be an exaggeration to say that you think all the world's problems can be solved if everyone were to read one particular book. He just means it would be helpful in one way or another. A lot of people feel that way about the Bible, of course, and there are a lot of people out there who are genuine evangelists trying to introduce others to the truths they have discovered in the New Testament.
For my part, I can think of only a few books that have made such a strong impression on me over the years that I have wanted, and maybe still want, to recommend them to others with evangelical enthusiasm. But my evangelical zeal has become somewhat diminished by the luke warm or negative feedback I have often received.
For example, I once recommended Raymond Carver to a middle-aged English woman, and she read one of his books of stories solely on my recommendation. (She was more the Thomas Hardy type.) Some time later when I asked her what she thought of Carver's book, she told me with narrowed eyes and gritted teeth, "I hated it!"
The books I remember as seeming to come as great revelations of one kind or another included:
What is Art? by Leo Tolstoy
The Catcher in the Rye by J. D. Salinger
Remembrance of Things Past by Marcel Proust
The Origin of Consciousness in the Breakdown of the Bicameral Mind by Julian Jaynes
The Selfish Gene by Richard Dawkins
New Grub Street by George Gissing
And I'm sure there were a few others that don't come to mind at the moment.
I don't usually recommend books anymore, and I'm not necessarily recommending these. The words "evangelical zeal" express the desire to share a moving experience, but they are also ironic because it is nearly impossible to communicate your feelings about books to others--just as others usually find it difficult or impossible even to get you to read something, much less appreciate something, that transported them. And your feelings often change with time.