It sounds like you're using a self-signed SSL certificate on the server, which is fine. I do the same thing for internal services. You'll just want to make your client aware of the custom SSL certificate as well, that way it knows that it can be trusted.
Something like this should do the trick:
req = Curl::Easy.http_post("https://mysite.com", loginJson) do |curl|
curl.headers['Content-type'] = 'application/json'
curl.cacert = "/path/to/ca.crt"
curl.cert = "/path/to/cert.pem"
end
Of course, you'll probably want to extract these string constants into a config file.