35 Then Jesus declared, “I am the bread of life. Whoever comes to me will never go hungry, and whoever believes in me will never be thirsty.36 But as I told you, you have seen me and still you do not believe.37 All those the Father gives me will come to me, and whoever comes to me I will never drive away.38 For I have come down from heaven not to do my will but to do the will of him who sent me.39 And this is the will of him who sent me, that I shall lose none of all those he has given me,but raise them up at the last day.40 For my Father’s will is that everyone who looks to the Son and believes in him shall have eternal life, and I will raise them up at the last day.”...
43 “Stop grumbling among yourselves,” Jesus answered.44 “No one can come to me unless the Father who sent me draws them, and I will raise them up at the last day.45 It is written in the Prophets: ‘They will all be taught by God.’ Everyone who has heard the Father and learned from him comes to me.46 No one has seen the Father except the one who is from God; only he has seen the Father.47 Very truly I tell you, the one who believes has eternal life.
The plain sense of 35-40 is that whosoever God sends, Jesus promises he will take care of them. In other words, don't worry, if God sends a person to Jesus, "I will lose none of those he has given me." They will partake of the resurrection of life and go on to have eternal life "I will raise them up." Verse 44 appears to be the issue for Calvinists, “No one can come to me unless the Father who sent me draws them." They want to say that it shows clearly that if you didn't come to Christ, God didn't draw you, and you can't come unless God draws you. God draws some people, "elects" them, and not others, and you can't do anything about it.
What seems wrong with this view is that it takes the particular and makes it universal. Jesus was saying to those present at that moment, that they weren't drawn. As in John 8:42-47, where he tells those present you are not of God, you are of the devil. That is, these people belong to a category which is outside the category of God's chosen people. In chapter 6, they belong to the uncalled category, in chapter 8, they belong to the category of those who's father is not the God of Israel. But neither of those categories are categories which belong outside of time. In other words, because people were not called to Jesus at the time of the discussion, they cannot respond to God at that time. The initiator must be God, and he hasn't initiated their calling before they spoke to Jesus. Similarly, in chapter 8, a person can become a "son of God," i.e. one who's father is God rather than the devil, by rejecting the evil one and joining with God and his people. Calling someone a son of the devil while they are engaged in evil acts is not necessarily a curse which excludes that person from ever participating in God's plan. Those who were formerly of the devil, can join to the body of believers through repentance and conversion, accepting the Gospel message. At that moment those people in Jesus' presence were not being called, and one has to be called before responding. It does not follow that these people or any others in their category will never be called (my father, a better theologian than me, believes that they may even be called at the time of judgement, i.e. after resurrection, but that's a whole other question). The question then is whether when a person is called, they can resist that grace. The answer seems to be clearly yes, that they must be willing. As in:
Revelation 22:17 (NIV)
17 The Spirit and the bride say, “Come!” And let the one who hears say, “Come!” Let the one who is thirsty come; and let the one who wishes take the free gift of the water of life.
Matthew 23:37-39 (NIV)
37 “Jerusalem, Jerusalem, you who kill the prophets and stone those sent to you, how often I have longed to gather your children together, as a hen gathers her chicks under her wings, and you were not willing.
God calls some people sometimes; he may or may not call all people at some particular time. When he calls, they must be willing to "take the free gift of the water of life." The picture that is drawn in John 6, is one of God initiating a calling of his people. Jesus is telling them that when they are called, don't worry, they can feel free to respond to that message because he will be faithful to them until the end. What this scripture excludes is a reading in which we initiate the calling ourselves. What it does not exclude is our choice once that calling has been made. As we see from Matthew and Revelation, the bible is replete with verses which assume a free-will response to that.