You could use ffmpeg to capture either of the mobile streams, (m3u8 streams), that tool I pointed to just uses Livestreamer. ffmpeg will give you the option of transcoding, eg. the audio stream to MP3, and then muxing into a MP4 container.
If you read a little more of the thread it also appears that adding the ngrp: and substituting _mobile is not guaranteed to get you the higher-res feed though.
This is only my interpretation of reading through the Stream Recorder forum but you could use rtmpserv or ffmpeg to capture the RTMP stream (the normal non-mobile stream) but you need to perform all the non-standard RTMP negotiation to get the URL to begin with.
MFC go out of their way to make capturing the RTMP stream hard - I think they also block the feed if more than one is detected going to the same IP with no valid login.
Just to clarify a bit more (possibly):
The URLs ending .m3u8 are the mobile streams, you can capture these using Livestreamer, ffmpeg, etc because it's just a HTTP streaming playlist. You can play them with VLC, MPC-HC, etc.
(It should be noted that there is no mobile (m3u8) stream for Private.)
The non-mobile stream uses Flash and the URL is something like RTMPS://....flv - that requires a Flash video player at the client end. rtmp(dump|suck|serv) and ffmpeg can capture RTMP(S) streams but there is handshaking to be done before the final URL is known. As far as I understand, MFC messed around with handshaking a little so that it's not standard, so making it a little harder to retrieve the final RTMP(S) URL.
As I said, this is my interpretation of what I've read.