They want her to convince John to confess. Both Hale and Danforth know an innocent man is going to hang. Hale quits the court because of it, but Danforth cannot. He needs John to confess in order to both spare his life and to save what little credibility Danforth\'s court has left.
They want her to convince her husband to sign the confession. He almost agrees, since she's doing the asking, but then they decide together not to kill the goodness in him by also taking away his dignity and the knowledge of knowing he died innocent without a lie of a confession hanging over his head.
Hale and Danforth do request Elizabeth to convince proctor, but she is not the soul person who convinced him. Befor, he was already thinking about confessing, b/c he realized that he is not a good man, and he cannot go like a saint, so diying like a saint would be a lie in itself. Two things that convince proctor not to confess is b/c proctor wanted to save his reputation... from his kids.. future... and the second is Rabecca.. who came in and was asstonished at proctors dicition, wich made him feel guilty