For starters the error "503 This mail server requires authentication when attempting to send to a non-local e-mail address." generally indicates that you need to be using UN/PW authentication to log into the SMTP server to send email outside your own domain.
"Your message did not reach some or all of the intended recipients." can be caused by a number of issues ranging from mail servers rejecting the message due to untrusted mail server on your end to mail delivery issues in general.
Overall it sounds like the mail server simply wants you to log in before sending mail to outside recipients and that should solve the issue so change authentication setting on your SMTP server setting to include your account username and password and you should be fine.
That being said... I fail to see how an issue with a 3rd party mail service not working in Outlook 2007 has anything at all to do with ASP.Net lol :P