Yes, the configure.php file needs to have the correct path to your servers:
If you are using a shared SSL Certificate as it sounds that you are:
you need to get the correct server path and enter it in you configure.php file.
It will be something like https://saturn.host.com/~username/page.html
If you have your own SSl cert, get with your host as to where these are required to be to function correctly as ZenCart only needs the correct server path and not the SSL cert location etc.
Code:
* URLs for your site will be built via:
* HTTP_SERVER plus DIR_WS_ADMIN or
* HTTPS_SERVER plus DIR_WS_HTTPS_ADMIN or
* HTTP_SERVER plus DIR_WS_CATALOG or
* HTTPS_SERVER plus DIR_WS_HTTPS_CATALOG
* ...depending on your system configuration settings
*/
define('HTTP_SERVER', 'http://yoursite.com');
define('HTTPS_SERVER', 'https://yoursite.com');
define('HTTP_CATALOG_SERVER', 'http://yoursite.com');
define('HTTPS_CATALOG_SERVER', 'https://yoursite.com');
I hope this points you in the right direction
You can also search this site for this subject.