curl: (35) Unknown SSL protocol error in connection
The main solution is to update curl and nss. If you are having an issue with curl through Apache/PHP you will need to restart PHP after.
It's important to remember that this error could mean a lot of things but most often it simply means that curl and openssl may be outdated and only allow newer secure ways of connecting to SSL.
In general here is how you would fix it in most cases:
yum -y update curl nss openssl
*Remember again if your PHP/Apache application is using curl it won't work until you restart Apache