Hi,
unless I am wrong in understanding your issue, it looks like the site you are trying to reach is performing a NTLM authentification request (Windows proprietary authentication method) ==> check in your log files some traces.
This methodology is really bad understood by squid (does requires "one to connect" method) and w$ is not fair in the story.
The workaround you can apply, unless squid community made great enhancement in last year, is to add this site in your browser "proxy exception" setting (or within your proxy.pac).
G.
PS: I have no idea about surrounding such issue with a transparent proxy setting : once the squid proxy has handle the request, it's too late ...