Your server is not compatible with modern secure communications.
Point your hosting company to
http://midwestglobalgroup.com/extras/curltester.php and ask them to fix the server's certificate chains and OpenSSL implementation.
That curltester.php file is a standalone file that implements basic secure CURL communications, which lets you point your hosting company to that file so they can't point fingers at Zen Cart. They can view exactly what that file is doing for communications, without passing the buck.
Properly securing/configuring the server is the hosting company's job. Only they have the necessary access to fix this simple misconfiguration problem.