"In the providence of God, Peter was permitted to close his ministry in Rome, where his imprisonment was ordered by the emperor Nero about the time of Paul's final arrest. Thus the two veteran apostles, who for many years had been widely separated in their labors, were to bear their last witness for Christ in the world's metropolis, and upon its soil to shed their blood as the seed of a vast harvest of saints and martyrs. AA 537.1"
"But ye shall receive power, after that the Holy Ghost is come upon you: and ye shall be witnesses unto me both in Jerusalem, and in all Judea, and in Samaria, and unto the uttermost part of the earth."
Both passages deal with witnessing to the world. The EGW passage describes Peter and Paul's final witness in Rome, fulfilling the call to witness to the 'uttermost part of the earth' mentioned in Acts 1:8.
"Ye have not chosen me, but I have chosen you, and ordained you, that ye should go and bring forth fruit, and that your fruit should remain: that whatsoever ye shall ask of the Father in my name, he may give it you."
The EGW passage speaks of Peter and Paul's ministry and martyrdom as bearing fruit. John 15:16 speaks of being chosen to 'bring forth fruit.' The connection is thematic, focusing on the idea of ordained ministry and its resulting impact.
"Since his reinstatement after his denial of Christ, Peter had unflinchingly braved danger and had shown a noble courage in preaching a crucified, risen, and ascended Saviour. As he lay in his cell he called to mind the words that Christ had spoken to him: “Verily, verily, I say unto thee, When thou wast young, thou girdedst thyself, and walkedst whither thou wouldest: but when thou shalt be old, thou shalt stretch forth thy hands, and another shall gird thee, and carry thee whither thou wouldest not.” John 21:18 . Thus Jesus had made known to the disciple the very manner of his death, and even foretold the stretching of his hands upon the cross. AA 537.2"
"Verily, verily, I say unto thee, When thou wast young, thou girdedst thyself, and walkedst whither thou wouldest: but when thou shalt be old, thou shalt stretch forth thy hands, and another shall gird thee, and carry thee whither thou wouldest not."
Ellen White directly references John 21:18 in this passage.
"Peter, as a Jew and a foreigner, was condemned to be scourged and crucified. In prospect of this fearful death, the apostle remembered his great sin in denying Jesus in the hour of His trial. Once so unready to acknowledge the cross, he now counted it a joy to yield up his life for the gospel, feeling only that, for him who had denied his Lord, to die in the same manner as his Master died was too great an honor. Peter had sincerely repented of that sin and had been forgiven by Christ, as is shown by the high commission given him to feed the sheep and lambs of the flock. But he could never forgive himself. Not even the thought of the agonies of the last terrible scene could lessen the bitterness of his sorrow and repentance. As a last favor he entreated his executioners that he might be nailed to the cross with his head downward. The request was granted, and in this manner died the great apostle Peter. AA 537.3"
"So when they had dined, Jesus saith to Simon Peter, Simon, son of Jonas, lovest thou me more than these? He saith unto him, Yea, Lord; thou knowest that I love thee. He saith unto him, Feed my lambs. He saith to him again the second time, Simon, son of Jonas, lovest thou me? He saith unto him, Yea, Lord; thou knowest that I love thee. He saith unto him, Feed my sheep. He saith unto him the third time, Simon, son of Jonas, lovest thou me? Peter was grieved because he said unto him the third time, Lovest thou me? And he said unto him, Lord, thou knowest all things; thou knowest that I love thee. Jesus saith unto him, Feed my sheep."
The EGW passage directly references Peter's denial of Christ and the subsequent commission to 'feed my sheep,' which is found in John 21:15-17. The EGW passage uses the biblical account to illustrate Peter's repentance and Christ's forgiveness.
"Then said Jesus unto his disciples, If any man will come after me, let him deny himself, and take up his cross, and follow me."
The EGW passage describes Peter's willingness to be crucified, even upside down, as a form of taking up his cross. This echoes the sentiment in Matthew 16:24, where Jesus calls his disciples to 'take up his cross, and follow me.'