I put together this video showing 5 different push button soft power latching circuits for the Arduino (works for other controllers too).
Soft Power Latching Circuits (https://youtu.be/7D9L9oS4AJM)
I bread boarded each one to show how it functions and I go through the schematic to show you how they work. Each circuit has its advantages and disadvantages.
Hopefully this can be a great resource for people looking to integrate something like this into their projects.