John did not carry the cross at all, although he was present at the Crucifixion - the only disciple who did not abandon Jesus. The cross was carried by Jesus himself, as was the custom, until he was so exhausted that he fell from the weight. Then we are told that a foreigner visiting Jerusalem, Simon of Cyrene, was commanded to carry it by the Roman guards there for the last stretch up to the Place of the Skull where Jesus was crucified.
Incidentally, when we hear of the 'cross' being carried, this was actually the cross-bar and not the whole cross. The cross-bar would have been carried across the shoulders so that the rough hewn wood would dig into the flayed and wounded back (caused by the savage whipping where chunks of flesh would be removed during the process) so that the maximum pain could be infllicted. Carrying (or dragging) a full cross like you see in Hollywood movies was not the practice of the Romans and never really happened. For one thing, crosses were usually in two parts - an upright and a crossbeam (which was carried), and secondly, just dragging a cross over one shoulder is much less exhausing and easier on a wounded back than carrying a crossbeam across the victim's shoulders, which would have been agony. The Romans were very clever people, wonderful architects, engineers, politician, artists and so on... but they were also astonishingly cruel.