Hydrogen bonding of chlorine
There is more to hydrogen bonding than electronegativity.  To get the strongest hydrogen bonds you need a compact atom to concentrate the negative charge for electrostatic attraction and to overlap well with the hydrogen 1s orbital for the molecular orbital/delocalized covalent bond contribution (yes, the latter is a thing.)  Nitrogen has that, chlorine not so much.  So, you get stronger hydrogen bonds with nitrogen than with chlorine.

Did you catch the wording?  Though chlorine has disadvantages it can form (weaker) hydrogen bonds under some conditions.  See here; the abstract indicates that the interaction can exist in some chlorinated organic compounds.