Bauckham, Richard. “Chapter 17, Polycrates and Irenaeus on John." Jesus and the Eyewitnesses: The Gospels as Eyewitness Testimony.” Grand Rapids: Wm. B. Eerdmans, 2006, pp. 438-471.
Bauckham considers Polycrates of Ephesus as the second best source of information we have about the author of John's Gospel (Bauckham 2006, 438). His statement, quoted by Eusebius, to Bishop Victor of Rome, which is the only extant writing of Polycrates, defends the Asian practice of calculating the date of Easter. Polycrates lists a number of people, claiming them as relatives. He includes among them John the apostle, also identifying him as the one "who leaned back on the Lord's breast," thus being the Beloved Discipole from John 21:24 (Bauckham 2006, 442). Bauckham further takes Polycrates' reference to John as martys to indicate that Polycrates considered him to have borne witness as a prisoner at Patmos, thus being the author if Revelation as well (Bauckham 2006, 443). His argument, further, is that the chronology in John's Gospel is superior to that of the Synoptics, which could be taken to represent the death of Jesus as the day before the Passover (Bauckham 2006, 444).
Bauckham considers what Polycrates meant by his reference to John "wearing the high-priestly frontlet" (Bauckham 2006, 445). After a review of the priestly customs, including Josephus' description of the wearing of the high priestly crown, Bauckham concludes this is an unambiguous statement that John served as high priest in the temple at some point. He provides several possible theories, among them that the John from Acts 4:6 is the son of Annas, the Beloved Disciple, and also the one referred to as Theophilus (Bauckham 2006, 449-451). This would, of course, require him not to be the son of Zebedee (Bauckham 2006, 452).
Irenaeus, cited in Eusebius, speaks of John the Evangelist in terms of the Beloved Disciple who leaned back on the breast of Jesus (Bauckham 2006, 454). Bauckham notes that Irenaeus, from Smyrna, was doubtless familiar with the views of the church in Ephesus, and that he had also learned from Polycarp before moving to Lyons, around 177 or 178. Bauckham observes that Irenaeus does tie his knowledge of Jesus to a chain of eyewitnesses - John to Polycarp to himself. The question of whether this John was the Son of Zebedee remains, however. Bauckham sums it up by saying, "What is revealing in itself is how difficult it is to find conclusive evidence one way or the other" (Bauckham 2006, 458).
Irenaeus does specifically refer to John the son of Zebedee five times, tying him "to his role in the Synoptic Gospels and Acts" and relating him closely with Peter and James (Bauckham 2006, 458). Bauckham does not find Irenaeus specifically equating the son of Zebedee with the Beloved Disciple. He further recognizes that early authors, including Irenaeus, will list people other than the Twelve as apostles (Bauckham 2006, 462).
Bauckham does find two apocryphal works which specifically identify the son of Zebedee as the author of the Gospel (Bauckham 2006, 463). The works are normally dated in the mid to late second century. Neither shows great familiarity with the traditions around Ephesus (Bauckham 2006, 464). Bauckham observes that Justin Martyr and Clement of Alexandria would serve as likely witnesses to the topic. However, their comments are not very hlepful (Bauckham 2006, 466-467).