Daniel, you might look at Testing TLS/SSL secured elastic cluster
Ii is about setup/configuration ES-7.8.0 fully x-pack secured cluster on RHEL with TLS/SSL on transport and http layers and tested it successfully with CURL and simple java client based on JEST API.
Just for clarity: Although Vadim's setup is definitely a good reference, the problem that Daniel has here is that CURL in his environment ( Windows ) is automatically sending client certificates even when it's not asked to, and unfortunately no amount of configuration on elasticsearch's side can fix that.
By the way. If instead of CURL you use web browser like IE or Chrome, everything works fine. You are prompted by the browser to select certificate for authentication of yourself to ES server and after that SSL handshake including verification of ES server DNS succeeds without any problems.
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