"in Galilee was gratifying to their ambition; they still hoped that He would give an evidence of His power which would lead the Pharisees to see that He was what He claimed to be. What if He were the Messiah, the Prince of Israel! They cherished this thought with proud satisfaction. DA 450.1"
"But we trusted that it had been he which should have redeemed Israel: and beside all this, to day is the third day since these things were done."
Both passages deal with the disciples' expectations of Jesus as the Messiah and their hopes for Him to redeem Israel. The EGW passage focuses on the disciples' ambition and desire for Jesus to prove His power, while Luke 24:21 expresses the disciples' disappointment after the crucifixion. They share the theme of messianic expectations.
"So anxious were they about this that they urged Christ to go to Jerusalem. “Depart hence,” they said, “and go into Judea, that Thy disciples also may see the works that Thou doest. For there is no man that doeth anything in secret, and he himself seeketh to be known openly. If Thou do these things, show Thyself to the world.” The “if” expressed doubt and unbelief. They attributed cowardice and weakness to Him. If He knew that He was the Messiah, why this strange reserve and inaction? If He really possessed such power, why not go boldly to Jerusalem, and assert His claims? Why not perform in Jerusalem the wonderful works reported of Him in Galilee? Do not hide in secluded provinces, they said, and perform your mighty works for the benefit of ignorant peasants and fishermen. Present yourself at the capital, win the support of the priests and rulers, and unite the nation in establishing the new kingdom. DA 450.2"
"For there is no man that doeth any thing in secret, and he himself seeketh to be known openly. If thou do these things, shew thyself to the world."
The EGW passage directly quotes John 7:4 and expands upon the context and motivations behind the brothers' words.