Kids cheat when they're not engaged with the material enough to learn it properly, or when the consequences for not cheating are too much for them to bear.
You gave them an assignment to do when you weren't actually teaching them, which means there's no way they can be properly engaged with the material. And you threatened their spring break with sitting in a room alone doing homework if they didn't get it done fast enough. You created a perfect breeding ground for cheating. Try creating an environment where kids don't feel that they need to cheat.
When I was in university I never heard of anyone cheating, because we were all treated like adults and we were engaging in material we liked. Try inspiring your students and treating them like adults. That means respecting their free time. If you don't give them respect as people, you won't get any respect as an authority.
How do you deal with students who say "my gut says it works this way. This is an easy problem, the answer is obvious. I don't know how to explain it to you any more simply"?
I mean, it takes 162 pages to formally prove that 1+1=2, but we got by just fine before we wrote down that proof. We just knew the answer, we couldn't explain how. If a student is gifted, a high school level problem could be as simple to them as 1+1 is to most people. They might know and not be able to explain how. Now, in a university environment I'd expect them to learn the proof, but that's not the point of high school maths, is it? The point of high school maths is to know how to solve the problem, not to know why the solution works.