に恥じない【にはじない】

に恥じない means "worthy of", "living up to": someone or something meets the standard of a name, a title, a reputation or an expectation. It follows that noun, usually な名, きたい期待, ひょうばん評判 or a role such as ぷろプロ, and it belongs to formal speech and writing.

はじる恥じる is "to be ashamed", so the pattern says the result gives no cause for shame when measured against the standard. It is a word of praise, or of resolve.

に恥じないように

With ようきごう（にきごう） the standard becomes a goal: "so as to live up to". This is how a speaker states a resolve.

に恥じぬ

にはじ恥じぬ uses the older negative ぬ and means the same. It is more literary, and common in speeches and slogans.