Does Apple push updates that require a restart within a set timeframe?
No, this is not Apple. If macOS has not been able to install updates without interruption, Apple software updates will show in Notification Center and will not force you to update.

The message in your question is from an MDM provider your organisation has installed on your computer, which in this case is JAMF. You should contact your organisation if you have any queries about the actions taken on the computer.