How do I access the Referer and User-Agent HTTP request header fields of the current HTML page from client-side JavaScript executed on that page?
Google Analytics manages to get the data via JavaScript that they have you embed in your pages, so it is definitely possible.
Related:
Accessing the web page's HTTP Headers in JavaScript
If you want to access referrer and user-agent, those are available to client-side Javascript, but not by accessing the headers directly.
To retrieve the referrer, use document.referrer.
To access the user-agent, use navigator.userAgent.
As others have indicated, the HTTP headers are not available, but you specifically asked about the referer and user-agent, which are available via Javascript.