The .NET HTTP client protocol handler is very picky about correct header formatting. I had similar trouble with a PIC based remote controller, also using authentication, that returned a response with only a single CRLF terminating the header section instead of two as the protocol specifies. Even after the firm tried putting the additional CRLF in for me it still failed so there was something else wrong as well. Browsers worked fine with it as they seem to be programmed to accept all sorts of malformed responses but the .NET HttpClient seems to require strict adherence to the specification. This might (or might not!) be your problem.
I worked around it using a Client from the Network library and rolled my own authentication as in the example attached. Luckily I only needed a single GET to send commands and get the information I needed back as a simple web page.