The box is designed to handle your DNS. At namecheap you’d setup glue records to denote the 2 nameservers internal to your box:
ns1.box.yourdomain.com
ns2.box.yourdomain.com
Once the DNS propogates, your box would be serving the appropriate values.
If you want to use namecheap as your DNS, and this is what you intend, my guess is you have the wrong values somewhere. You’d capture the correct values to use under the System → External DNS page on your admin screen. Specifically, it’s the record that looks something like:
mail._domainkey.yourdomain.com
On your other question. Yes, you can setup an A record in the Customer DNS section of the admin screen. The A record should point to the IP of your host/website. Then the box will serve the right DNS for your website and visitors will be seeing the content hosted somewhere other than the box.