I am not using a API but I have been looking into the API like a mad man trying to find some relevance to it. I am opening the website and downloading all of it at the moment to
Make a call to get the user /users/*user-id*
and compare the result for non-existent users.
http://instagram.com/developer/endpoints/users/#get_users
You could also use the users search /users/search
call to search by username, and iterate through the results and see if any match the username you are looking for exactly. If none of the results match exactly, then you know they don't exist (or are for some reason hidden).