It sounds like perhaps Multi-Account Containers is the feature that you are looking for. Although it's not full Private Browsing mode, you can set specific websites to always open in a new container, which will keep the cookies in that container separate from the general Firefox browsing.

However, it's important to note that (unlike Private Browsing mode), websites visited in Multi-Account Containers will still be recorded in your browsing history, as will content downloaded within these containers.

Hope this helps.

no this is not but thanks. We are looking to make sure when a users goes to a specific website that its open in a private browser so their logins are not cached. Cant use it linked as a shortcut as all the users would have to do is open a tab and login and then their login gets saved. there used to be this open that worked great. Chrome has a great item but we are looking for this for FF and IE now. I dont know why this really isn't a thing that should be built in with the way privacy is today. I would like to save sites that would just open in private when i type that site in the browser then it opens in private.

What about this extension?

https://addons.mozilla.org/firefox/addon/privatebookmark/