The red color means very low charge. If it's flashing red while connected, it's possible that shuffle's battery has become worn out and it is no longer able hold a charge.
If it still appears in the iTunes sidebar (under DEVICES) when it is connected, you can try doing a Restore. With the shuffle selected in the sidebar, go to the Summary tab over to the right. Click the Restore button. This will erase the shuffle, re-install its software, and set it to default settings. If the cause of this problem is related to software (or settings), the Restore should fix it. Otherwise, it is probably a hardware issue.
Which iPod model is it? If it's been less than one year since the purchase date, it is under warranty; you should contact Apple tech support either at a Genius Bar (if there's a nearby Apple Store) or by using the Contact Us link at the bottom right corner of this web page.
The oldest iPod shuffle second gen is about 6 years old. But that's still pretty old for a shuffle. Lithium batteries are not designed to last forever.
It will cost you more than the $50 for a new 4th gen shuffle to get it repaired, assuming it needs a new battery. There are online guides to do it yourself, for example
if you're handy with doing such things. You can get the part (including tools) for under $10 on eBay.