Is the biblical idea of being "born again" compatible with eastern religious ideas of reincarnation? Just a simple question: is the biblical idea of being born again compatible with eastern religious ideas of reincarnation?
Being "Born again" and being reincarnated are completely different concepts.

The concept of being born again comes from this Bible passage below.  I know it's long but the theological and doctrinal implications found in this passage are profound and critical to understanding what it means to be "born again" and what salvation requires.


  John 3:3-21 (NIV)
  3 Jesus replied, “Very truly I tell you, no one can see the kingdom of God unless they are born again.” 
  
  4 “How can someone be born when they are old?” Nicodemus asked. “Surely they cannot enter a second time into their mother’s womb to be born!” 
  
  5 Jesus answered, “Very truly I tell you, no one can enter the kingdom of God unless they are born of water and the Spirit. 6 Flesh gives birth to flesh, but the Spirit gives birth to spirit. 7 You should not be surprised at my saying, ‘You must be born again.’ 8 The wind blows wherever it pleases. You hear its sound, but you cannot tell where it comes from or where it is going. So it is with everyone born of the Spirit.”
  
  9 “How can this be?” Nicodemus asked. 
  
  10 “You are Israel’s teacher,” said Jesus, “and do you not understand these things? 11 Very truly I tell you, we speak of what we know, and we testify to what we have seen, but still you people do not accept our testimony. 12 I have spoken to you of earthly things and you do not believe; how then will you believe if I speak of heavenly things? 13 No one has ever gone into heaven except the one who came from heaven—the Son of Man. 14 Just as Moses lifted up the snake in the wilderness, so the Son of Man must be lifted up, 15 that everyone who believes may have eternal life in him.”
  
  16 For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life. 17 For God did not send his Son into the world to condemn the world, but to save the world through him. 18 Whoever believes in him is not condemned, but whoever does not believe stands condemned already because they have not believed in the name of God’s one and only Son. 19 This is the verdict: Light has come into the world, but people loved darkness instead of light because their deeds were evil. 20 Everyone who does evil hates the light, and will not come into the light for fear that their deeds will be exposed. 21 But whoever lives by the truth comes into the light, so that it may be seen plainly that what they have done has been done in the sight of God.


TL;DR
If you balked at reading all of that, it basically says this:


You must be born again, which means born of water (from your mother) and born of the Spirit (from God)
In order to be born of the Spirit, you must believe in Jesus
You must live your life according to Jesus and follow him in order to truly believe in him.


That's the essence of being "born again".