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.