First of all your store didn't even load up in my browser. I get redirected to Yahoo Search.

Second. Either you are not providing enough information on here or your host is the worse ever.

For using any SSL you will need a dedicated IP. So by them telling you, they can't install dedicated IP on the server it means you need a different host right away.

I would suggest finding a host in the Zen Cart certified host link above. Hostmonster works good for small website. They won't charge you for installing your SSL. It takes a couple of minutes to do and if you know what you are talking about you can't talk your way out of paying it with any host.

So to sum it up.

SSL is very important to any online business.
Dedicated IP is a must (You have to pay for it, usually a couple of dollars a month).
Zen Cart desn't have a way to assign secure URL to checkout page. The CMS is smart and configured to display secure URLs only when needed on important pages only where you collect information from customers.
Your host is bad and you need to run away. Moving your cart isn't that hard to do, if you follow the information on this forum you can do it yourself.