When you get the red x thing that means it has failed at the lowest levels, the hardware has actually gone down. This is different than when you get software issue and messages like limited connection.
There are not a lot of things that can cause this. You could have some rare driver issue for the ethernet port but these chips are not complex so you seldom even find driver updates. The most common cause is a bad ethernet cable. It is the cheapest thing to try to replace. Other than that you have either a bad port in your router or in your PC. Neither of those is actually repairable so you never want to have that problem.
I would buy a new cat5e cable from a reputable vendor (no crappy cheap CCA or flat cables from amazaon).