Doesn't Snapchat mainly rely upon the iOS or Android platform having some software that prevents screen shots if a 'no screen shots' flag is set? I always thought this was their core defense.
However, the binary level protection is only tangentially anti-screenshot insofar as it also blocks third-party 'scraper' apps. Fundamentally, it's about requiring use of the Snapchat app to access the Snapchat APIs, and as we know from the ongoing saga with other social networks like Twitter, this is ultimately about business control over posts (preventing spam-bots, scheduled clients, and so on) and at the end of the day, ad revenue.
1. Open the Snapchat app for a bit so it is downloaded / cached.
2. Turn on Airplane mode.
3. Screenshot and screen record to your desire.
4. Delete the whole app.
5. Turn off Airplane mode and download the app again.