By this point in the play, Reverend Hale has had a complete turn-around, and is desperately trying to reverse some of the arrests that he has made. Because he signed so many death warrants at the beginning of the chaos, he feels that he is personally responsible for their deaths; after he realizes that the courts are corrupt and the girls are false, he understands that these people are going to die in vain. He doesn't want people dying just because some silly girls accused them, and he signed their death warrants. So in Act Four he is going to each person slated to hang, and pleading with them to confess in order to save their lives.
Hale goes to Elizabeth, because he knows that John won't confess; he is hoping that Elizabeth can plead with him, and soften his heart. He knows that John is stubborn and proud, but if his own wife will get in there and beg for him to live, maybe he will confess and save his life. So, Hale begs Elizabeth to get John to confess. So far they haven't because they do not want to lie (confessing to being a witch would be a lie), so he tells her that
"it may well be God damns a liar less than he that throws his life away for pride."
This is the argument that Elizabeth says is the devil's. Her reasoning is that lying is bad, no matter what, and it would be just like the devil to try to convince people it was okay to lie, because it was less bad than being prideful. It is taking something that they consider a sin, and making it look okay by comparison, and that's a sneaky trick of the devil to get people to do bad things. So that's what she meant.
I hope that those thoughts helped clear things up a bit; good luck!
By Act IV, Rev. Hale is sick of the way things have gone in the Salem witch trials. This is why he wants John Proctor to make a confession even if that confession is false. He says that there is no way that God wants anyone to give up their life for a principle because life is the most precious thing there is.
I think that Elizabeth says this is the Devil's argument because of the idea that the Devil can twist the truth to make it seem to say the opposite of what it really is. I think she feels that Hale is trying to twist the truth to get John to do something wrong just because Hale is ashamed of his own role in the trials.
You should note that Hale himself says that he has come to do the Devil's work by trying to get good people to lie.