Does John Proctor care about his 'name'? Why does he refuse to put his name publicly on the document knowing that he will hang as a result?
One's good name and reputation were a lot more important in 17th-century Salem than they are today. John Proctor's whole identity as a man and as a Christian is bound up with that of the wider community. He's anxious, then, to do everything he can to maintain his standing in the community as a good Christian man. This explains why he tears up his signed confession in anger:
"I have given you my soul; now leave me my name!"
John would rather die than live in shame and ignominy. He is fighting to protect the good name of the Proctors, not just that of himself and his present family but also of his ancestors and descendants yet to be born. The community of Salem is all John knows. It's given him everything; without it he has no social identity. Individual death is preferable to the social death he will suffer if his signed confession is made public for all to see.
Yes, John Proctor does indeed care about his name. Near the end of the play, he is asked to sign a document confessing to witchcraft. Although he is willing to admit to the crimes verbally as a way to end the insanity within the village of Salem, he knows that signing his name will ruin not only him but his family for generations to come. The signed confession would only propel the "witch hunt" into the future and justify the actions and cover-up that has transpired to this point. Ultimately, Proctor realizes that the only way to put an end to the vicious cycle is to refuse to sign the confession, and therefore protect his name, knowing full well that he will hang because of his refusal. His sacrifice is one of truth and justice.
In the play The Crucible, John Proctor's name is a symbol of him as a person and the life he ahs lived. He does not initially recognize how important his honesty is but over time he begins to realize that he can not stand by and allow his wife to die. His wife tells him not to go forward about the girl and tell the truth because she does not want her children left alone.
When John is told to sign the papers he realizes that his signature would be an allegation against many innocent people and he can not do that so he refuses to sign and demonstrates that he is a good and just man.