Can you play Minecraft for Xbox One without having Xbox Live?
Solution 1:
You can't really even use the Xbox One without an account signed in (you can really only do basic stuff, but once you try to do something it will ask you to sign in), so the first time they played, they were logged in. I'm guessing when you turned the system off it signed out.
You want to go to the circled tile and select a profile to sign in.
Account doesn't mean Xbox Live. You don't need Live to play, but you do need an account/profile
Solution 2:
I was having the same issue, but I figured it out. Yes, you can play without a membership. All you need to do is, when creating a world, uncheck the box that says "ONLINE" under the Difficulty setting.