The only way to check if a post is not there anymore on Facebook is to search for it with a User Access Token of the User who posted it. Every Object on Facebook gets a specific ID, you only have to check if that ID still exists. If not, you get an Error from the API.
For example: https://developers.facebook.com/tools/explorer/?method=GET&path=10203433018378479&version=v2.0
The path parameter is the ID of the Post.
Keep in mind that you need the read_stream
permission for that, and you need to let Facebook approve it for other users or it will only work for Admins/Devs of your App. It is not very likely that you will get the permission approved for this though. It usually only gets approved for Apps on "Platforms without a native Facebook experience".
Edit: My bad, i was thinking about User posts, but your question was about Pages. In that case, all you need is an App Access Token (App-ID|App-Secret). The API Call would be the same, you just need to know the Post ID.
About Access Tokens:
For getting the feed of a Facebook Page, see the Facebook docs (including code samples): https://developers.facebook.com/docs/graph-api/reference/v2.0/page/feed/