Currently, there is no in game way of saving highlights in Overwatch. This is something that's been hinted at by Blizzard that they will be implementing in the future, but until then you are going to have to use third party software in order to actually save the video highlights to your own computer. Otherwise, any time you log out all of your highlights will be gone. If you are using Windows 10, which by now I think a lot of us have been forced into, or soon will be, it actually has a built in feature to record game play.
Step 1)Open the Game Bar in Windows 10
Simply press the windows key and g at the same time while you are playing a game and it should open right up for you. If you don't see it, and you are running windows 10, then you may need to be running the game in windowed mode instead of fullscreen to see it.
Step 2)Watch your highlight video in game
Go to your recent highlights and set it up to play the video you are looking to record to your computer. Once it is set to play, you'll need to set up to record it at the same time.
Step 3)Hit record!
Just hit the red record button in the game bar as soon as you press play on the highlight video. You can also use the keyboard shortcut of the windows key + Alt + R to either start or stop recording.
That's it, now you can locate and play those highlight reels any time by locating them in your user account’s Videos\Captures folder.
Step 1)Open the Game Bar in Windows 10
Simply press the windows key and g at the same time while you are playing a game and it should open right up for you. If you don't see it, and you are running windows 10, then you may need to be running the game in windowed mode instead of fullscreen to see it.
Step 2)Watch your highlight video in game
Go to your recent highlights and set it up to play the video you are looking to record to your computer. Once it is set to play, you'll need to set up to record it at the same time.
Step 3)Hit record!
Just hit the red record button in the game bar as soon as you press play on the highlight video. You can also use the keyboard shortcut of the windows key + Alt + R to either start or stop recording.
That's it, now you can locate and play those highlight reels any time by locating them in your user account’s Videos\Captures folder.