Jesus' words in John 6:35 are critical for understanding the rest of the chapter - "I am the bread of life. He who comes to Me shall never hunger, and he who believes in Me shall never thirst" (cf. 4:14). Jesus plainly affirms that He is the bread from heaven. The verbs are in the present tense, so the meaning is this: He who comes and continues to come to Jesus and he who believes and continues to believe in Jesus will never be in want spiritually. Such a person will satisfy his spiritual hunger and quench his spiritual thirst (cf. Matt. 4:4). Jesus, the bread and water of life, can fully satisfy the deepest needs of the soul. Man may search for fulfillment elsewhere (e.g., money, possessions, careers, family, entertainment, etc.) but will ultimately be left empty without heavenly food and drink for the spirit. Additionally, one must keep on eating and drinking in order to live. This is true both physically and spiritually.
When these Jews requested a greater sign (cf. 6:30) they were admitting what Jesus stated in John 6:36 - "You have seen Me, and yet do not believe." They had already seen much more than just a sign, they had seen Him. That is, they had witnessed the marvelous miracle from the day before (and probably many others on previous occasions) and they had listened to Him teach, yet they remained unconvinced. It wasn't that they hadn't been shown adequate evidence. They didn't believe because they didn't want to believe. Thus, the situation was hopeless. For example, if a person refuses to believe in the sun that our planet orbits when he can see its light, feel its heat, and witness its life-giving power, then what more can one do to demonstrate to him the existence of the sun?
These Jews had just asked for bread (cf. 6:34), yet they reject the spiritual bread (Jesus) that God is trying to give them! Had He offered them some more barley bread it is certain they would have accepted it, but instead He offered them that which is infinitely better, yet they rejected it (i.e., Him).
Jesus declared in John 6:37 - "All that the Father gives Me will come to Me" (cf. 6:44,45). There is nothing here or anywhere else in the Scriptures that indicates that the Father arbitrarily chooses or rejects certain ones. God wants all to be saved by coming to the knowledge of the truth, believing it, and obeying it (cf. I Tim. 2:4; II Pet. 3:9). The Father certainly did not prevent those present from coming to have faith in Jesus. Jesus' point here is that the Father gives Him those who believe in Him. The reason why many do not believe and are not given by God to come to Christ is simply because they refuse to believe (cf. Matt. 13:14,15)!
"The one who comes to Me I will by no means cast out" (John 6:37). Regardless of a person's past or position in life, Jesus does not reject those who come to Him in faith. However, this is not to say that a child of God cannot fall away and be eternally lost (cf. Gal. 5:4; James 5:19,20; II Pet. 2:20-22). If one comes to Jesus in faith (and continues coming in faith), then he will be faithfully obedient. Such a one will never be rejected by Jesus.