I just received a Yubikey 4 Nano replacement for my old, broken nano and noticed the blinking immediately. Using the information in this post I was able to use the NEO Manager to disable the CCID protocol to stop the blinking.
However, I'm a loss to understand why the blinking is necessary in the first place. In fact I just discovered that my Yubikey could be used to store GPG keys/sub-keys. This would come in extremely handy in daily use, except for the fact that enabling CCID causes my Yubikey to blink every couple of seconds, nonstop. The blinking is just out of my sight and unbearable. I've turned off CCID mode to retain my sanity.
If this blinking behaviour is on purpose and unstoppable (while using the CCID protocol) then my opinion of Yubikey just got downgraded. I still like the product but will no longer shout its praises from the rooftop (or my office) and that makes me sad.