You do not need to be online to read an eBook downloaded to your PC from B&N.
Hover over the book in Nook for PC, it should show a menu box and it should say either read now, or download. If it says read now, the book is on your PC and you don't need internet access to read it. If is says download, click that so the book downloads and then again, you don't need internet access to read it. In fact, I'm not sure how you read it online.