You can put it under the Privacy category, and it can just be a simple checkbox "Clear cookies on close", with a popup warning the user that this could be a dangerous action. Something like "Are you sure? THis will clear all cookies and log you out of all your accounts when you quit Orion".
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Automatically clear browsing data on exit
But, I still think using private windows is a better idea.
Where is the setting, how does it look, how does it work.
Chrome
Chrome places the feature in Privacy and Security > Cookies and other site data. Here, you can enable it and set exceptions.
This is the popup for adding exceptions. Key thing here is that it allows for specific subdomains and catch-alls.
Firefox
Firefox places their feature in a similar position in their Privacy & Security section. They also have a manage exceptions area, which functions pretty much the same as Chrome's.
4 months later
Are there ay updates on this feature? It's been a few months now...
5 months later
I'd also like to add that it should look similar to the "remove history" items. e.g. "after one day" "after one week". 
5 months later
- Edited
Merged 17 posts from Delete cookies on browser close.
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