I am creating a secure web based API that uses HTTPS; however, if I allow the users to configure it (include sending password) using a query string will this also be secure or should I force it to be done via a POST?
Yes, it is. But using GET for sensitive data is a bad idea for several reasons:
Therefore, even though Querystring is secured it's not recommended to transfer sensitive data over querystring.
[1] Although I need to note that RFC states that browser should not send referrers from HTTPS to HTTP. But that doesn't mean a bad 3rd party browser toolbar or an external image/flash from an HTTPS site won't leak it.