This excellent play concerns itself with, amongst other things, the themes of respect and reputation. For Hale, having gone through his own personal journey from avid witch hunter in Act I to disillusioned cynic in Act IV, somebody's name or reputation is not worth dying for, and any lie is better to save yourself than speaking truth and being hung for it. Hale's comments therefore say a lot about his character and the change that he has experienced.
However, before answering your question, we need to understand where Proctor is coming from and why he makes the decision that he does. He is condemned to hang along with two other women who are refusing to "confess" to crimes they did not commit. In addition, during the course of the play, he has had to confront his own failings and sin in his adulterous relationship with Abigail and the way that he has not been the upright and respected man that others in Salem think of him as being. This decision therefore comes at a very important point in his life. Note how he justifies his refusal to give the confession to Danforth:
Because it is my name! Because I cannot have another in my life! Because I lie and sign myself to lies! Because I am not worth the dust on the feet of them that hang! How may I live without my name? I have given you my soul; leave me my name!
For Proctor, any life he would have gone on to live after confessing falsely would not have been worth living, as he would have been permanently stricken by his guilt and shame. He has the opportunity to do the right thing, and he desperately needs to make the right decision in order to be able to live with himself, however short the rest of his life will be. Therefore, personally, I do not think Hale was correct. For Proctor, bearing in mind his character and circumstances, this was the right decision.