When is the “last day” that is spoken of in John 6:39, John 6:40, John 6:44, and John 6:54? To me, it appears that it is the Rapture that he is referring to. But in John 11:24, Martha is talking about Jesus raising up Lazarus (a Jew) at the “last day”. The only other time the exact words “last day” (not “last days”) is mentioned is John 7:37: ‘On the last day of the feast, the great day, Jesus stood up and cried out, “If anyone thirsts, let him come to me and drink.” Is it possible that all of these verses could be referring to the Rapture that will take place on the last day of the Feast of Tabernacles (year unknown)?
Jesus never taught directly on the rapture, although having learned of it from Paul’s teaching we can see Him hinting about it in various places. I think the references you cited from John 6 and John 11 concern the resurrection of the believing dead. For those who became members of the Church this resurrection will take place at the time of the rapture (1 Thes. 4:16-17). According to Daniel 12:2 the resurrection of Old Testament believers will take place at the time of the 2nd Coming. Jesus could have been referring to either one or both of these events.
I don’t see any connection between the last day of the Feast of Tabernacles in John 7:37 and the Lord’s references to the last day in John 6 and 11. Different prepositions are used in connection with the phrase and they tell the story. In John 6 and 11 Jesus said the resurrection will take place “at the last day” meaning it will happen in the end times, but not necessarily on the very last day of them. In John 7 the phrase is “on the last day” indicating that Jesus made His remarks on the very last day of the feast.