Specifically, what evidence do we have, apart from the Gospels, that any of Jesus’ disciples actually had visions of Jesus after his death?
Dr. Bart D. Ehrman:
The short answer is that apart from the Gospels the only other good evidence we have are the book of Acts and the letters of Paul
Steefen (personal essayist), author of Insights on the Exodus, King David, and Jesus / The Greatest Bible Study in Historical Accuracy, 2nd Ed.:
He was the Messiah. And when, upon the accusation of the principal men among us, Pilate had condemned him to a cross, * * those who had first come to love him did not cease. He appeared to them * * spending a third day restored to life, – First century historian, Joseph ben Matthias (Josephus) contemporary to James, brother of Jesus.
Note: The Testimonium Flavianum is authentic. Scholars who said it was not gave reasons. One reason was that it was out of context, unrelated to the passage following it. That simply is not true. The Testimonium Flavianum is followed by a passage of a god appearing to a woman and spending time with her – god made man and walking among us, a claim of Christianity. Furthermore, this god appeared to her on the third day. As we know, Josephus changed camps from Jewish rebel leader against Rome to supporting the Romans against the Jewish rebels. IN CHARACTER, Josephus tells us the God who appeared to the lovingly devout woman and spent time with her revealed that he was not god.
This is Josephus’ stance of atheism against his former religion which was warranted because after the Babylonian exile, after the Seleucids, after Rome wipes away the success of the Macabbees, after Rome takes away a Jewish king in Jerusalem, after the garments of the high priests have to be given to Romans, and after the handwriting on the wall that Rome was going to win the Jewish-Roman war, after Joseph ben Matthias (Josephus) gave his speech with tears in his eyes and speaking (while crying) to the rebels, explaining if they did not stop, we would lose the Temple to destruction, after all this and more, zealotry for the Jewish God was not amounting to anything but demise.
The TF is not just Antiquities of the Jews, Book 18, Chapter 3, Section 3 but Sections 3 and 4! (This will appear in the 2nd ed of my book and will be on my website in a few weeks: http://www.waterbearingfish.com/SaulJosephus_ben_Matthias.html
The second reason the TF is authentic is because while Vespasian enriched Josephus for his support of the Romans, Vespasian also enriched another rabbi, Rabbi Johanan ben Zakkai who headed the university of Yavne after the Jewish Revolt was put down. Rabbi Joseph ben Matthias and Rabbi Johanan proclaimed the Christian story. We know of the TF. As for Rabbi Johanan, he proclaims the impregnation of Mary by God, the Lion of Judah, and that God tried to save his people through her son.