When I use TwiML to send commands, Twilio API (TwilioProxy/1.1) makes requests to my Apache server on HTTPS without SNI support.
I need to block non-SNI clients (
I've just had a similar problem. Twilio was working fine with SNI (CloudFlare free Universal SSL) up to end April 2015, but suddenly stopped working.
Twilio support says that the TwilioProxy doesn't yet support SNI. This surprises me since it was working with SNI until recently. They say SNI support is on the backlog and they 'hope to support it in the future'.
A workaround may be to put CloudFlare Pro in front of your server for non-SNI, but this is currently $20 for the first site and $5 for additional sites.
Alternatively, other alternatives (e.g. Plivo) may support SNI. I'm investigating them; they're cheaper than Twilio too but don't have TwiML.
To be clear, Twilio has supported SNI since August 2015 which came almost 3 months after the original answer David provided. In the new Console - Account Settings you will find that here: https://www.twilio.com/console/account/settings