Geocortex Essentials will make a direct connection to the URL you specify in the map service connection. It's not going to use your browser here, so even if the connection works from the browser you may have to perform additional configuration to have the connection work with Essentials.
The first step will be to ensure that Essentials trusts the certificate used by ArcGIS Server. If you make a direct connection on port 6443, then the SSL certificate used may be the default self-signed certificate, which is not signed by a trusted authority and also may not be suitable for the domain name of the server.
You can either import the SSL Certificate into the Trusted Certificate Store for the server, or you can switch to using an SSL certificate that is issued by a known Certification Authority and that is valid for the URL. We recommend the latter, in combination with the web adaptor, as this allows you to use:
as the URL for the service.
Note: In this case it is a good idea to explicitly specify the token URL. If it's not specified, Essentials will try to infer it from the map service URL. However, it will first attempt to use an SSL version of the URL to obtain the token (since we'll be sending a username and password over the network). If that fails for any reason then we'll blacklist the SSL address and switch to the non-SSL version. If regular HTTP is not allowed then we will not be able to obtain a token and the site will fail completely.