My guess would be wrong file location but that doesn't make sense if it works on Windows.
Make sure there are no weird characters or spaces in the url or file names.
Also, try setting the folder's permission settings to 755 on your server. I've had external media issues with that before.
Do you have a link to share with us?
Flash player debug version spits out this error:
SecurityError: Error #2148: SWF file http://quietly.fastmail.net/media/bfly.swf cannot access local resource \\VBOXSVR\Documents\Work\Cengage\HTML 5 Illustrated\Unit K\Author 1st\SV working unit files\media\bfly.m4v. Only local-with-filesystem and trusted local SWF files may access local resources.
This tells me you probably have the file located in the wrong location. Make sure when you are testing that you are testing the live version, not local.
You are pointing to a video on a local hard disk. When you are linking the video in Flash, make sure the path is relative to the swf file and make sure your server mirrors that. Organize your files and folders carefully.
Thanks for your help. I hadn't realized that Flash would embed absolute rather than relative paths. I fixed that and got everything happy.