Q: IPod Shuffle Not Charging

It should at least charge, even if iTunes is NOT installed.  That's just a function of getting power from the USB port (like connecting it to a power adapter).  So reinstalling iTunes probably will not help.

Do this to Reset the shuffle

http://support.apple.com/kb/HT1655

If you are connecting it to a USB hub, try connecting it directly to a USB port on the computer.  If that does not help, try shutting down (powering off) the computer and disconnecting all USB peripherals.  Start it up with only standard keyboard and mouse connected (if used).  Run iTunes.  Connect shuffle to a direct USB port (it needs to be at least USB 2.0).  If possible, test the port with something similar, like a USB flash drive.

If that does not help (and you confirmed the port is working), the possibilities are bad USB cable or something wrong on the iPod.

During the first 90 days after purchase, you are entitled to "complimentary phone support" from Apple tech support, so you should contact them; you can use the Contact Us link at the bottom right corner of this page.  Or, if there is an Apple Store nearby, you can make a reservation at the Genius Bar and take your shuffle to Apple tech support there.  Either way, you can see about getting it replaced, if it's defective.