Here the requirement for saving our lives is losing our lives, which means denying ourselves, dying to ourselves, and following Jesus. Letting the old proud self die is not works. It's the death of works. That's who's dying. When that old rebel dies, I don't earn anything.
Nobody boasts in having to be put to death on a cross.
You don't earn anything by getting crucified. Following Jesus is required, but it is not works. Following Jesus is no more working for Jesus than the pigeons of Elliot Park are working for me when they follow me around because I have the bread crumbs they want. Following Jesus would be working for our salvation if we were following him to meet his need and not for ours. If you die you live.