I have bought an IPod shuffle 4th gen and it won't come up in Itunes and it won't turn on, in Itunes it just had a red light and when I unplugg it, no light at all.
Did it previously work with your computer and iTunes? Or is this brand new, never previously connected to your computer?
You can try this procedure to reset the iPod shuffle
Then, shut down your computer (power off) and start it up, with iPod shuffle NOT connected. Run iTunes. Then, try connecting the iPod shuffle again.
If that iPod is less than one year old, it is under warranty. You can contact Apple (or make an appointment at an Apple Store Genius Bar) to see about getting it replaced (if it's broken).