Summary: The reader of John's Gospel should find little suprising in this passage. Jesus brings together lots of themes and words he has used before. One could bridge "sending away and staying" or "seeking and finding" in some neat ways. The granddaddy phrase though is that faith is a "work of God"; we must return to Greek 101 for some translation help here.
Two word pairs very common in John's Gospel
Seek and find [ζητουντες & ευροντες (seek and find; 25 and 24)] From the very beginning of John's Gospel to the end, Jesus asks people what they are seeking (original disciples and Mary Magdalene after the resurrection). Jesus is constantly being sought too. (If you look up the word, it appears nearly every chapter). Likewise, people are finding Jesus (Nathaniel in John 1 and Peter finding fish and discovering Jesus in John 21). Yet Jesus is also good at avoiding detection. Always sought; sometimes found.
I have not explored this fully, but I think one could argue, quite well, that Jesus only is found when he chooses to let himself be found, when he takes the first step, for example, by calling the disciple's name.
Send away and stay [αποστειλεν (from αποστελλω, "send" 29) and μενω ("abides", 27)] One cannot say enough about the importance of these two concepts in John's Gospel. We could put them together and say that in Jesus Christ, we will be still yet conquer the world. This is a powerful image of a Christian, one who is moved yet finally unmovable in the core. I could be more pithy, but it is late at night. Sermonize away...
απολλυμενην ("destroy" or "perish", 27) This is a strong word used in the Bible at key points. Jesus says if you want to gain your life, you must "lose" it; Paul says that the message of the cross is foolishness to those who are perishing. The Bible certainly sets up a strong contrast between life in and outside of Jesus.
Grammar review: subjective genitive
το εργον του θεου (29)
We could translate this genitive in a number of ways: "The work done by God" or "the work which belongs to God" or "the work which is offered to God." You could probably squeeze most theological arguments into how we understand faith -- is it a work for God or a work from God. I vote with the later one generally, and definitely in this case, where the whole emphasis is on Jesus, the true bread, coming from God.
Grammar review: ου μη
ου μη (35)
This is the strongest denial possible in Greek.