"Many of the Israelites saw no help in the remedy which Heaven had appointed. The dead and dying were all around them, and they knew that, without divine aid, their own fate was certain; but they continued to lament their wounds, their pains, their sure death, until their strength was gone, and their eyes were glazed, when they might have had instant healing. If we are conscious of our needs, we should not devote all our powers to mourning over them. While we realize our helpless condition without Christ, we are not to yield to discouragement, but rely upon the merits of a crucified and risen Saviour. Look and live. Jesus has pledged His word; He will save all who come unto Him. Though millions who need to be healed will reject His offered mercy, not one who trusts in His merits will be left to perish. PP 432.1"
"Looking unto Jesus the author and finisher of our faith; who for the joy that was set before him endured the cross, despising the shame, and is set down at the right hand of the throne of God."
Both passages encourage looking to Jesus/the remedy provided by God, rather than dwelling on one's own problems or weaknesses. The focus is on faith and reliance on Christ for salvation/healing.
"God is our refuge and strength, a very present help in trouble."
Both passages deal with trusting in God for help in times of trouble. The EGW passage uses the imagery of the Israelites looking at the bronze serpent to be healed, and applies it to looking to Christ for salvation. The Psalm speaks of God as a refuge and help. The connection is strong thematically.
"Many are unwilling to accept of Christ until the whole mystery of the plan of salvation shall be made plain to them. They refuse the look of faith, although they see that thousands have looked, and have felt the efficacy of looking, to the cross of Christ. Many wander in the mazes of philosophy, in search of reasons and evidence which they will never find, while they reject the evidence which God has been pleased to give. They refuse to walk in the light of the Sun of Righteousness, until the reason of its shining shall be explained. All who persist in this course will fail to come to a knowledge of the truth. God will never remove every occasion for doubt. He gives sufficient evidence on which to base faith, and if this is not accepted, the mind is left in darkness. If those who were bitten by the serpents had stopped to doubt and question before they would consent to look, they would have perished. It is our duty, first, to look; and the look of faith will give us life. PP 432.2"
"For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life."
Both passages emphasize the importance of faith and belief for salvation/healing. The EGW passage uses the example of the bronze serpent to illustrate the need to 'look' in faith, while John 3:16 states that 'whosoever believeth' will have everlasting life.