Your browser will not send Windows credentials to the web-application unless it believes that the URL is on the Intranet. When this is not the case it will prompt you for credentials via a simple login-dialogbox.

If you on the offending computer add the URL of your webapplication under Internet Option/Local Intranet/Sites it will work. But this is an It-problem. Get IT to make the organizations computers see your URL as being on the Intranet.