Shinydrive fully supports using an SSL (Secure Sockets Layer) configuration. There is nothing you need to configure in Shinydrive for this to work, it's all done on the Tomcat level. Since there is nothing to configure in Shinydrive, you will need to look at Apache Tomcat's documentation for setting this up. There are guides available based on the version of Tomcat you are using from Apache.
After you have switched to SSL, you will need to make sure the Shinydrive client is using https:// in the server URL field and the new port number (Tomcat's default SSL port is 8443).