"Now it was the Feast of Dedication in Jerusalem, and it was winter. And Jesus walked in the temple, in Solomon's porch. Then the Jews surrounded Him and said to Him, 'How long do You keep us in doubt? If You are the Christ, tell us plainly.' Jesus answered them, 'I told you, and you do not believe. The works that I do in My Father's name, they bear witness of Me. But you do not believe, because you are not of My sheep, as I said to you. My sheep hear My voice, and I know them, and they follow Me. And I give them eternal life, and they shall never perish; neither shall anyone snatch them out of My hand. My Father, who has given them to Me, is greater than all; and no one is able to snatch them out of My Father's hand. I and My Father are one."
The Feast of Dedication, held in late December, was a yearly eight-day commemoration of the cleansing and rededication of the temple that had been polluted by Antiochus Epiphanes in 167 B.C. Its observance was not restricted to Jerusalem but was kept wherever Jews lived throughout Palestine. It was a national holiday comparable to July 4th in America and was celebrated with joy and strong feelings of patriotism. This feast is known today as Hanukkah.
Since this feast celebrated national deliverance it is possible that some of the rulers saw this feast as an opportune time to tempt Jesus to verbally declare Himself to be the Messiah or deliverer from the current Roman oppression. It is likely that they surrounded Him with the intention of detaining Him until He answered them to their satisfaction - "How long do You keep us in doubt? If You are the Christ, tell us plainly." Their motive is certainly not curiosity. They desire Jesus to make a clear statement that they can use as an accusation against Him.
Jesus affirmed that He had told them and they did not believe (e.g., John 8:58). He had previously revealed His divine nature to them, but they had disbelieved and had become enraged. They had not believed Him then, nor would they believe Him now even if He gave them a plain verbal statement.
"The works that I do in My Father's name, they bear witness of Me" (10:25; cf. 5:36). Had the Jews properly understood the Old Testament concept of the Messiah, then they would have understood Jesus' claims. However, they were looking for a physical savior and did not see Jesus as such. Their unbelief was not due to a lack of proof but to their own stubbornness. They simply did not want to believe though the evidence was abundant.
Because they did not believe, they were not His sheep. It is not the case that they could not believe. God has not made any decree rendering anyone incapable of believing. By choosing not to believe they demonstrated that they were not His sheep. They had stubbornly resisted Jesus' teachings and miracles time and again, choosing to remain blind in their traditions.
"My sheep hear My voice, and I know them, and they follow Me" (John 10:27). The ones that can be called Jesus' sheep are the ones that hear (i.e., heed) His voice. Anyone can become a sheep of Jesus if he will listen to and follow the Lord (cf. Rom. 10:17; James 1:22). Those who hear and respond in complete obedience are the ones that God knows (i.e., approves of).
"And I give them eternal life, and they shall never perish; neither shall anyone snatch them out of My hand" - A careful study of John 5:24 (as well as Mark 10:30 and Titus 1:2) will lead one to the conclusion that eternal life is only received in this life as a promise. Jesus will actually give eternal life to the faithful at the resurrection and such will never lose their souls (i.e., "perish") in the hereafter. However, those who do not hear His voice and follow Him will suffer forever in hell.
Although Jesus clearly states that no one is powerful enough to snatch a sheep out of His hand, He does not exclude the possibility of a sheep wandering away from the Lord of his own free will (cf. Rom. 11:22; I Cor. 9:27; II Pet. 2:20-22; etc.). Those who forsake the Lord can blame no one but themselves for their straying (cf. I Cor. 10:13; James 1:13,14). So, while it is true that no one can be taken from God against his will, one may choose to leave God and the Lord will not stop him.
No one is able to snatch a sheep away from Jesus for the Father has given the sheep to Him, along with the power to keep them safe from harm. This is the case because the Father is "greater than all", and thus His power is absolute and fully sufficient to protect His own.
"I and My Father are one" (John 10:30). Specifically, Jesus and the Father are one in purpose (i.e., the protection of faithful followers, in this context). Jesus' actions were in perfect harmony with the will of the heavenly Father. In general, They are one in essence (i.e., They possess the divine nature equally, yet They are distinct as members of the Godhead).