A direct "like" button should not be possible, since Facebook needs to sandbox the whole thing. Both like button formats (iframe and xfbml) use a sandboxed iframe, so nobody but facebook can set "likes" or know anything about the user.
For that to work with Flash, I think facebook would have to provide the button itself as a SWF file so you can load it in Flash. Then it can create a sandbox within itself and thus load the user information safely. Very much like the Chromeless Youtube Player, where it's almost impossible for Flash to access the actual FLV URL (even through introspection).
But even so, I don't think this would provide enough security... realize that you are dealing with very private user information and credentials here.
The only way I know you can "like" stuff through Flash is first doing the whole Facebook Connect thing (several popup windows asking the user for permissions) and then using the API of your choice to do the actual "liking", for instance, with the Graph API:
http://developers.facebook.com/docs/api#publishing