Could not seem to find a way to go straight from BHO to my pc with a video posted here by a member. But if it originated on Youtube, then you can click the name "Youtube" at the bottom right of the video to jump to the same video on the Youtube internet channel. Then you can grab the video with Free Youtube Downloader.
I could be mistaken, but I think all videos on the Hangout reside on YouTube. No videos are actually stored or hosted on the Hangout. So having to go to YouTube to make any downloader works makes sense.
Not sure which browser you're using but I use the CCleaner Browser which has a down loader built in. you just hover around the top right of the video and pick the resolution you want and click download.