Speaking generally first, no - there is no part of specification or browser implementations that supports arbitrarily clearing all cookies from a particular domain through the use of http GET parameters. This would be a horrible security hole if it did exist in either specification or practice. If it was possible, I could maliciously redirect your browser to some known site and destroy information that you did not want destroyed.
Now to your particular situation ... I gather that what you want to do is not so much clear the cookies from your own browser in this particular site, but rather, you are trying to implement a "clear cart" button on your own BigCommerce site. Is that right? If so, cookies is probably not the path forward that you are seeking. I don't think BigCommerce exposes shopping cart functions through their api. But I would be interested to hear from the BigCommerce folks on that. Your best bet to add a "clear cart" function is to add javascript to the page that iterates all cart items and makes that http call you mentioned. If you do this, make sure to have some graceful fallback for the case where they change the uri. :)