My first question is do these
I have been fighting a similar issue with SSL handshake issues for the past few days. After reboots and upgrades for GHOST, we started seeing errors like this in our error logs constantly:

*579 SSL_do_handshake() failed (SSL: error:140A1175:SSL routines:SSL_BYTES_TO_CIPHER_LIST:inappropriate fallback) while SSL handshaking,

in conjunction with an elevated error rate in client requests to nginx in the initial connection phase. I'm not completely sure if the two issues are correlated to be honest, I'm still in the troubleshooting process.

I am on a Debian Wheezy system and it started happening with the libssl package 1.0.1e-2+deb7u13 and continues with u14. As soon as I rolled back libssl to u12 and restarted nginx, the logging of errors goes away. I then tested ssl to make sure we weren't vulnerable to POODLE or Heartbleed, and it's all clear. I would recommend trying to go back a few versions in libssl, restarting nginx and see if that helps, making sure you're not leaving yourself open to the major vulnerabilities.