Reverend Hale is burdened by the fact that before he realized how ridiculous all of the arrests and condemnations were, he signed a lot of death warrants. He says, "There is blood on my head!", and so, in an effort to assauge his guilt, he is trying to keep more people from dying. The way to do that was to get them to confess. He reasons with Elizabeth saying, "It may well be God damns a liar less than he that throws his life away for pride." He is trying to convince her that lying to save your life is worse than dying because pride won't let you lie.
Elizabeth merely wants John to make his own decision: "As you will, I would have it." But she does admit that "I want you living, John. That's sure." I imagine she is greatly struggling with her desire for him to live and her desire for him to do what is right. She feels that him not confessing so far "speak goodness" in John, and she called Hale's reasoning "the devil's argument", so she obviously feels that not confessing is the right thing to do, but she wants John to find his own way. When he does, and tears up his confession, she says, "He have his goodness now," which for her, is a great statement of love and trust in his integrity.