This is a puzzling one indeed. I’ve never seen it before, but here are perhaps a few things you could try.
Did you check the security permissions on the included .js files? Perhaps there is a security setting on one of those files that would prevent just that one user from accessing the underlying JavaScript. I don’t expect that this would be it, because if it was, I would expect there to be either JavaScript errors (which you said you did not have) on the page or a login prompt. But, it may be worth checking anyway.
Another thing you could try is checking the user’s JavaScript settings under Internet Options in IE. That is one thing that could be different for that user account. The textboxes are all created by calls to JavaScript functions, so perhaps JavaScript calls are disabled?
Finally, look through the HTML source code of the page, as well as your style-sheet file for any references to ms-formbody. This class is used for all text box fields. Perhaps there is something causing the class to be hidden.
I know none of these offer solutions, but maybe you can at least find some clues from these ideas.