Hi,
Actually, with "Connection: closed", the server is supposed to close the connection immediately after sending the data, and that should be handled by HttpResponse without issues.
However, it's possible that the server closes the connection in a wrong way. For example, when HTTPS is used, the connection should be closed by sending the closenotify alert message, because a lack of that would open the connection to truncation man-in-the-middle attacks, where the client could be tricked into accepting incomplete data without being able to detect that. So perhaps the lack of closenotify is actually the issue with SugarSync?
Please post the error message you get, or rather a debug log - that should help us to figure out what's actually going on, and hopefully suggest a workaround for that particular issue.