Is there a way to check if a twitter username exists? Without being authenticated with OAuth or the twitter basic authentication?
You can try to grab the http://twitter.com/username page and read the response to see if you get the "Sorry, that page doesn’t exist!" page.
As @Pablo Fernandez mentioned in a comment, it will be better (faster, more reliable) to check the response header, which will be "404 not-found" if the user doesn't exist.