I am trying to accept all certificates, and/or accept self-signed certificates using Apache HTTPClient version 4.5 (tutorial link here)
One of the above approaches should work in case of self-signed certificates, but the weird thing is you are getting same exception in all the approaches.
I feel during SSL session establishment or handshaking protocol is not being accepted either by client or by server.
The best solution here is to debug the application.
In case of tomcat, add -Djavax.net.debug=all in setenv.sh or setenv.bat files and then restart the server.
Or you can follow this tutorial.
The OP just needed to change the port when connecting to SSL:
//For HTTPS
HttpHost httpstarget = new HttpHost("mysite.com", 443, "https");
//For HTTP
HttpHost httptarget = new HttpHost("mysite.com", 80, "http");