The claim is false...
Strictly speaking, the claim is easily proven false by searching a scripture index of the writings of the Ante-Nicene Fathers. While this particular index seems imperfect, it does show that many verses (particularly from the shorter letters) are not quoted or even referenced.
We can also take advantage of the Philip Schaff translations of the Fathers, which are available online and use the KJV for scripture quotations. John 3:16 turns up in a full-text search as does 1 John 3:16. However, we wouldn't have any record of 3 John 3. (Sadly the letter ends at verse 14 so I can't be consistent in my survey.)
There is a complete index of Biblical texts of the 10-volume Ante-Nicene Fathers, which shows a single verse from John's third epistle (3 John 11), which is found in volume IV, page 63. Here, Tertullian writes that when it comes to marriage, we should not follow the examples of David and Solomon (who married many women), but to "follow the better things" (Joseph, Moses, and Aaron who married just once). But the quote fits just as well with 1 Peter 3:11 or Psalm 37:27 as it does with 3 John 11.
... but much of the Bible was quoted.
On the other hand, we find the early church did quote the New Testament extensively. Augustine, for instance, gave extensive lectures on the Gospel of John and the First Epistle of John, which were written down. They serve as a very comprehensive commentary on the two books from which we can extract nearly the entire text (in Latin). Even better, he preached the text in order, so we can follow along. His other expository sermons on the gospels probably cover most of the text of those books.
Additionally, allusions to the New Testament permeates the earliest Christian writing (both Orthodox and Gnostic). The index of New Testament references in Clement of Rome, Mathetes, Polycarp, Ignatius, Barnabas, Papias, Justin Martyr, and Irenaeus shows they (as a group) were familiar with nearly every canonical work. Not all of these are direct quotes, but demonstrate close familiarity with the texts.
One of the central principles of the Historical Method is the more independent sources that are available. the more confidence we have in the truth of a historical claim. When it comes to the text of the New Testament, there is already a remarkable number of manuscript copies. With the exception of variations, we know what the text said. While we have no need of the early fathers to be confident in the content of the New Testament, their quotations of it give us information about its canonicity.