The principal arguments used against John's authorship of the Gospel of John are:
- Contradictions in the accounts
- The Gospel is written in Greek
- The Gospel was written too late to have been written by a close eyewitness of Jesus' ministry
This has already been addressed in SLM's answer. It should be noted, however, that the Tubingen school who popularized the claim that the Gospels were fantasy, fiction, and forgery, was operating from a very significant premise: they assumed naturalism in advance (no such thing as the supernatural).
Since the existence of the supernatural is the very subject under consideration in the Gospels, to assume naturalism in advance is a textbook case of begging the question.
If perceived contradiction is evidence against an eyewitness account, that would rule out not only most Biblical history, but almost all multiply attested accounts of all of human history. J. Warner Wallace explores this reality in Cold Case Christianity, showing that eyewitness accounts--real, genuine eyewitness accounts--usually do differ in detail, focus, and sequence. It's when you have identical accounts from different eyewitnesses that you should be suspicious of collusion and/or fabrication.
In any event, as it relates to the date of Jesus' crucifixion, the preponderance of the evidence supports 14 Nisan, as reported by John. Whether or not the Synoptics got the right date is a separate matter, but John's chronology checks out. See my work on the subject in this video series: Chronology in the Life of Jesus.
2. Could John have written a Gospel in Greek?
This is a bit of a bait-and-switch argument. Could John the son of Zebedee have written a document like this in Greek himself? Probably not.
But that's not how writing worked in the first century--when people wanted to write something formal, they generally used an amanuensis (a scribe). See further relevant discussion in this post on Biblical Hermeneutics Stack Exchange.
Whether or not John could write Greek is irrelevant--he wouldn't have been the one doing the writing. In antiquity the "author" was not the person who scratched the symbols onto a scroll--that was the scribe--the author was the person who decided what was going to be said, and under whose authority the final draft was approved and published.
(for responses to arguments suggesting Jesus, John, and members of their social class could not speak Greek, see my video What Languages Did Jesus Speak?. Spoiler: Whether or not they could write it, Jesus & John could definitely speak Greek)
Some claim that the Gospel of John was written too late to have been written by John the apostle. This argument is circular; I explore it in detail in this video.
Skeptical 19th century scholars deliberately assigned late dates to the New Testament documents to discredit the possibility of eyewitness/informed authorship. Modern skeptics who are unaware of this history will cite these 19th century dogmas in order to claim they are disproving eyewitness authorship, when in reality they've just argued in a circle. Lack of eyewitness authorship had to be assumed in advance to get the late dates in the first place.
In any event, we have multiple, early attestation that John the son of Zebedee was still actively ministering in Ephesus at the time of the accession of Emperor Trajan, in AD 98--more than 60 years after Easter. Even if the Gospel of John was written near the end of the 1st century (as many historians hold), John was still around at this time (see esp. Irenaeus Against Heresies 3.3.4).
Arguments that John couldn't have written the Gospel of John are popular because they carry implications that many find desirable. However, these arguments don't get any stronger by frequency of repetition.