Yes it should. Most likely the form element names are wrong somehow. That would explain the checkbox not working as well as the values not appearing in the database. 
The Check box works, it just does not auto fill the shipping fields...
<input type="checkbox" name="eC_SameAs" id="eC_SameAs" value="1" onclick="eCart_copyBillingToShipping(this);">
Here's a sample of the first field for first name in shipping:
<input type="text" name="shipping_firstname" id="shipping_firstname" value="">
as opposed to the billing field:
<input type="text" name="firstname" id="firstname" value="">


