The particular Controller action will only respond to a POST request, this is by design.
In Firefox, Safari and Chrome, the response is still being returned, with the appropriate HTTP 200 code and JSONP content, the only difference is that the XmlHttpRequest object being used by JQuery is setting the status code to 0, and the responseText to empty.
Originally, I thought this was due to COR HTTP Preflighting (Http Access Control), whereby a custom header or a content-type other than text/plain would cause an additional HTTP request (with an OPTIONS) verb to be sent to the server. I can see in Fiddler2 that the OPTIONS request is being responded to with a HTTP 404.
The web server is IIS7 (but the production web server will be an IIS6 box). In IIS7, I can see the standard OPTIONSVerbHandler listed in the handlers, but I'm not convinced this is actually doing anything (in fact, I can't even find any documentation about the OPTIONSVerbHandler anywhere).
To get round this, I modifed the JQuery library to not set the custom header, and change the content-type to text/plain instead of application/json, and Firefox finally starts bypassing the OPTIONS request, and just plain POSTs.
The problem still lies in an empty response (according to the XmlHttpRequest object), even though Fiddler2 shows that a successful HTTP 200 response, with content is being returned.