I would write The Directorate confirms than none of the persons in the apartment have any health issues, nor do they have any criminal or administrative convictions.
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AnonymousThe Directorate confirms than none of the persons in the apartment have any health issues,I must admit I hadn't heard of "administrative convictions" before!and neithernor do they have any criminal or administrative convictions.
tamguatlayThe Directorate confirms than none of the persons in the apartment have any health issues, neither do they have any criminal or administrative convictions.That's OK too.
tamguatlayI think it is all right to retain "neither" in the OP's sentence.Yes, but with a semi-colon or "and".