"“These things said He: and after that He saith unto them, Our friend Lazarus sleepeth; but I go that I may awake him out of sleep.” “Our friend Lazarus sleepeth.” How touching the words! how full of sympathy! In the thought of the peril their Master was about to incur by going to Jerusalem, the disciples had almost forgotten the bereaved family at Bethany. But not so Christ. The disciples felt rebuked. They had been disappointed because Christ did not respond more promptly to the message. They had been tempted to think that He had not the tender love for Lazarus and his sisters that they had thought He had, or He would have hastened back with the messenger. But the words, “Our friend Lazarus sleepeth,” awakened right feelings in their minds. They were convinced that Christ had not forgotten His suffering friends. DA 527.2"
"These things said he: and after that he saith unto them, Our friend Lazarus sleepeth; but I go, that I may awake him out of sleep."
The EGW passage directly quotes and comments on John 11:11, making it a clear and direct allusion.
"“Then said His disciples, Lord, if he sleep, he shall do well. Howbeit Jesus spake of his death: but they thought that He had spoken of taking of rest in sleep.” Christ represents death as a sleep to His believing children. Their life is hid with Christ in God, and until the last trump shall sound those who die will sleep in Him. DA 527.3"
"For if we believe that Jesus died and rose again, even so them also which sleep in Jesus will God bring with him."
Both passages present death as a form of sleep for believers in Christ. The EGW passage expands on this concept, connecting it to the idea of life being hidden with Christ and the resurrection at the last trump.
"“Then said Jesus unto them plainly, Lazarus is dead. And I am glad for your sakes that I was not there, to the intent ye may believe; nevertheless let us go unto him.” Thomas could see nothing but death in store for his Master if he went to Judea; but he girded up his spirit, and said to the other disciples, “Let us also go, that we may die with Him.” He knew the hatred of the Jews toward Christ. It was their purpose to compass His death, but this purpose had not succeeded, because some of His allotted time still remained. During this time Jesus had the guardianship of heavenly angels; and even in the regions of Judea, where the rabbis were plotting how they might take Him and put Him to death, no harm could come to Him. DA 527.4"
"Then said Jesus unto them plainly, Lazarus is dead."
The EGW passage directly quotes John 11:14, making the connection explicit.