Hmmmm, I've not uploaded HTML before, so I cannot say if that's where your problem lies. I do know that KDP and Pubit use completely different conversion software, so where HTML works for KDP it might be why it doesnt want to work so well with Pubit. You might want to try to save your WIP as a 97-2003 doc (as it plays nicer with Pubit's conversion) and see how that goes, or since you seem well versed, convert your HTML to epub and upload that instead.
Although there are many similarities between the html files used to create Kindle and Nook ebooks, there are some surprising differences that can create the kind of trouble you and your customer are observing.
Sadly, it sounds like you didn't test your Nook ebook before publishing it. There's a very good reason why PubIt! provides an epub of your converted book for you to download and test before you publish it. Without testing it, you are bound to have trouble if you are new to the ePub standards upon which the Nook platform is based.
Personally, I would NEVER leave an ebook for sale on ANY e-reader platform until I walk in my customer's shoes by being my first customer, buy a copy, download it to a clean e-reader and test it. You are treating your customers poorly if you don't carefully test your self-published ebooks before someone else buys one.
Regarding your problem ...
I ran into a similar-sounding problem the first time I converted an ebook to the Nook platform. The book in question had already been converted to the ePub format for KindleGen. The mobi file I produced worked great on all Kindle devices and software. But it crashed on Nook devices and software. In my case, the problem was that I used a single html file for the text of the book. It was a large book (335,000 words) and the html file was 4 MB (4,000 KB). That was the problem.
I had ignored the B&N PubIt! ePub Formatting Guide which stated that html files should be less than 300 KB because the guide is old (from 2010). I was mistaken—the 300 KB limit still stands. As soon as I divided the text into smaller single-chapter files, all problems disappeared. On the Nook, the smaller files also makes navigating the ebook much faster.
If your html file is 300 KB or larger, that would be the first problem to solve: divide it into smaller files. This also means you will no longer be able to submit your book as an "html" file because PubIt! requires an ebook to be submitted as a single file. Therefore, I also strongly recommend that you go all the way, as Melissa suggested, and create an epub file, yourself.
If you are good with html, there is nothing to stop you from creating an epub file. You'll need to prepare a opf and ncx file (both in xml) and a css file. Then you zip them and rename the zip file with an ".epub" file name extension. It's fairly easy for someone with basic html skills. And all of the work you do will help you submit better ebooks to Amazon KDP, also. So none of this will be wasted effort.
Once you have your book converted to an epub file, use the free Google ePub validation tool to test your file. Don't submit it to PubIt! until it passes the validator with zero errors. (Note: The oversize html files will not generate an error so you'll want to adhere to the Nook peculiarities listed in B&N's guide such as html file size.)
To read the details of my ordeal and the solution see the topic "Advanced problem solving of a large ePub for the Nook ..." in this forum. The ebook in question was my wife's and, because I was new here and wasn't sure I would need to open my own account for this forum, I wrote under her PatMH account and noted the same in my sig that it was me writing for her.
I hope the problem you've discovered is nothing more than the html file size. But there are other things that can cause the symptoms you described and we probably can't provide any further help without looking at your html code.
Kind regards, David