If it's been less than one year since purchase, and it is defective, you can get a replacement under warranty.
I haven't had it that long..
So, is it the current 4th gen iPod shuffle? This article shows the four types.
If you have not already, try restarting your computer. Do this to Reset the shuffle
Run iTunes and try connecting it again. Make sure the cable is fully connected to the iPod's headphones jack. Use a direct USB port (not a hub) and try more than one port.