"The mission of Christ in this world was not to gratify idle curiosity. He came to heal the brokenhearted. Could He have spoken any word to heal the bruises of sin-sick souls, He would not have kept silent. But He had no words for those who would but trample the truth under their unholy feet. DA 730.3"
"But he answered and said, It is not meet to take the children’s bread, and to cast it to dogs."
Both passages illustrate the principle that truth should not be given to those who will reject or abuse it. Jesus' silence before Herod is presented as an example of this principle, similar to the warning in Matthew 15:26.
"Christ might have spoken words to Herod that would have pierced the ears of the hardened king. He might have stricken him with fear and trembling by laying before him the full iniquity of his life, and the horror of his approaching doom. But Christ's silence was the severest rebuke that He could have given. Herod had rejected the truth spoken to him by the greatest of the prophets, and no other message was he to receive. Not a word had the Majesty of heaven for him. That ear that had ever been open to human woe, had no room for Herod's commands. Those eyes that had ever rested upon the penitent sinner in pitying, forgiving love had no look to bestow upon Herod. Those lips that had uttered the most impressive truth, that in tones of tenderest entreaty had pleaded with the most sinful and the most degraded, were closed to the haughty king who felt no need of a Saviour. DA 730.4"
"Give not that which is holy unto the dogs, neither cast ye your pearls before swine, lest they trample them under their feet, and turn again and rend you."
The EGW passage directly applies the principle of not casting pearls before swine (Matthew 7:6) to Jesus' silence before Herod. Herod's rejection of previous prophetic messages makes him unworthy to receive further revelation.
"Herod's face grew dark with passion. Turning to the multitude, he angrily denounced Jesus as an impostor. Then to Christ he said, If You will give no evidence of Your claim, I will deliver You up to the soldiers and the people. They may succeed in making You speak. If You are an impostor, death at their hands is only what You merit; if You are the Son of God, save Yourself by working a miracle. DA 730.5"
"He saved others; himself he cannot save. If he be the King of Israel, let him now come down from the cross, and we will believe him."
Both passages describe a demand for Jesus to perform a miracle to prove his divinity, although the specific context and actors differ. The underlying theme of challenging Jesus to prove himself is present in both.