John Eglinton is actually the pseudonym of William Kirkpatrick Magee, and you'll notice that throughout "Scylla and Charybdis," Stephen sometimes refers to him by his actual name. Both in the novel as in life, he was an accomplished essayist and an influential figure on the Dublin literary scene. W.B. Yeats even called him "our one Irish critic." In 1904, when the novel is set, Eglinton was the assistant librarian at the National Library.
Eglinton frequently demonstrates his learning, and treats Stephen a bit condescendingly throughout their discussion on Shakespeare. Whereas Stephen is a fan of Aristotle, Eglinton much prefers Plato, which puts him in agreement with the mystic George Russell. Eglinton is also similar to Russell in doubting whether or not a man's biography is relevant to his work. Throughout, it is very clear that Stephen is trying to impress Eglinton and not the other way around.
Stephen gradually wins Eglinton over, and by the end of the discussion, it is clear that Eglinton is looking at him with respect. Yet Eglinton cannot understand why Stephen would waste all of their time with a theory that he himself does not believe. He doubts that he can help Stephen publish it if even the author is not committed to it.