I suspect this is related to the the way Safari sandboxes media playback. It seems like the the page authenticates but the video file is treated as a totally separate request requiring its own login. That second authentication request fails on certain configurations and the transfer hangs waiting for credentials.
The only solution I've found, which is less than ideal, is to specifically exempt mp4 files from authentication. Add the following to the site's .htaccess file and mp4 files will be playable without logging in.
Keep in mind that if someone knew the exact path of the video files they would be able to view the files without logging in.
# Exempt movies from password protection to prevent extra login prompts
<Files ~ "\.(mp4|m4v)">
AddType video/mp4 mp4
order allow,deny
allow from all
satisfy any
</Files>
Note: The tilde tells Apache to match at the end of the filename, not in the home directory.
I was able to recreate this behavior on Lion (10.7.5) with Safari 6.02. Mountain Lion (10.8.2) with Safari 6.02 presented a second authentication window and played the videos correctly.
The only other thing, which I doubt, is that the server mime-types need to be explicitly set for mp4 playback. The second rule of the .htaccess block takes care of this, just in case.