Re: "I was using Photo Booth to record a video. While the video was still recording, I clicked Command-Q on my keyboard to quit. Photo Booth closed. When I reopened Photo Booth, the video was not saved. Is there any way to recover this video? Is there any way to recover this video?"
Unfortunately, It seems the video is gone for good. Quitting without stopping the recording allowed no time / action for saving it.
(An accidentally deleted video could be recovered while PhotoBooth was still open using Edit > Undo Delete..)
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Re:Is there any way to enable autosave upon quitting?
Sorry I don't know for sure, but I really don't think so.
Photos are auto-saved after the shutter clicks, videos when recording is stopped.
It is always good practice to save a back-up copy of anything valuable, preferably before quitting an app.
(Eg: A flash drive is a cheap and easy option, saves Mac's storage, and is handy for recovering a file quickly.)
Photo Booth has both export and share options available. (You can "share" a photo or video to yourself )
Apple Support has a PhotoBooth User Guide online, which details all the software's possibilities.
For details: You can click the "Table of Contents" at the top of the Welcome page that opens from the link below:
Photo Booth User Guide for Mac - Apple Support