Is it permissible to cite the Hadith: "Every nation has a character, and the character of my nation is modesty. And indeed, if Allah Almighty wills to destroy a servant, He removes modesty from him, and when He removes modesty from him, you will not meet him except as one detested"?

There is no hadith with the wording: "Every nation has a character, and the character of my nation is modesty (haya')." The correct narration is what Ibn Majah and others narrated with the wording: "Indeed, every religion has a character, and the character of Islam is modesty (haya')," and it is an authentic hadith.

As for the hadith: "Indeed, when Allah عز وجل intends to destroy a servant, He removes modesty from him..." it is very weak, its chain of narration is fabricated, and it is not permissible to attribute it to the Prophet, peace and blessings be upon him, because that would be a lie against him. There are many authentic hadiths about the virtue of modesty that suffice over weak hadiths.