I’ve received this error several times now, so I am posting the workaround so I do not have to continually look this up. Credit to http://www.zjulian.com/mitigating-burp-suite-error-burpsuite-handshake-alert-unrecognized_name.
Start Burp Suite with the following command:
C:\Path_to\Burp Suite Pro> java -Djsse.enableSNIExtension=false -jar burpsuite_pro_v1.7.06.jar
Here are the technical details that I pulled from the other link:
Server Name Indication (SNI) is an extension to the TLS protocol that indicates what hostname the client is attempting to connect to at the start of the handshaking process. This allows a server to present multiple certificates on the same IP address and port number and hence allows multiple secure (HTTPS) websites (or any other Service over TLS) to be served off the same IP address without requiring all those sites to use the same certificate. It is the conceptual equivalent to HTTP/1.1 virtual hosting for HTTPS.
Certain misconfigured websites will send an “Unrecognized Name” warning in the SSL handshake, causing Java to mishandle the connection.