I have written up a procedure for integrating SSL into Arduino sketches on Galileo. I'm using wolfSSL specifically for this, but there's no reason why the process couldn't be adapted to use OpenSSL instead. It's not for the faint-of-heart, but if you know your way around Linux you shouldn't have any trouble with it. I did my testing on a Galileo 2 board but this should work on the original Galileo as well. It worked on both the SD-Card Linux image and the image from the IoT Developer's Kit (though the latter has an extra hoop that you need to jump through).