SSL 3.0 is considered insecure as it is vulnerable to the POODLE attack that affects all block ciphers in SSL. TLS 1.0 does include a means by which a TLS implementation could downgrade the connection to SSL 3.0, thus weakening security.
To disable TLS1.0 while enable TLS1.1 and TLS1.2 in an Cpanel environment, place the following in /usr/local/apache/conf/includes/pre_main_global.conf file:
SSLHonorCipherOrder On SSLProtocol -All +TLSv1.1 +TLSv1.2
Test shows the handshake error for TLS1.0, which means TLS 1.0 has been successfully disabled on port 443. While TLS1.1 and TLS1.2 were enabled successfully.
openssl s_client -connect example.com:443 -tls1 openssl s_client -connect example.com:443 -tls1_1 openssl s_client -connect example.com:443 -tls1_2
example output:
#openssl s_client -connect 127.0.0.1:443 -tls1
...
SSL-Session:
Protocol : TLSv1
Cipher : 0000
...
this mean TLS 1.0 not supported
#openssl s_client -connect 127.0.0.1:443 -tls1_1
...
SSL-Session:
Protocol : TLSv1.1
Cipher : ECDHE-RSA-RC4-SHA
...
this mean TLS 1.1 is supported
You might run some further tests on other ports and TLS1.0 seems fine with them:
openssl s_client -connect example.com:465 -tls1 openssl s_client -connect example.com:993 -tls1 openssl s_client -connect example.com:995 -tls1 openssl s_client -connect example.com:2078 -tls1 openssl s_client -connect example.com:2083 -tls1 openssl s_client -connect example.com:2087 -tls1 openssl s_client -connect example.com:2096 -tls1
Actually, those ports are owned by different processes and are not managed by a web server. You will have to configure the corresponding service to disable TLS1.0 for those ports.