To form a hydrogen bond, what must the non-hydrogen atom (NN, OO, or FF) involved in the bond possess?

It can occur with any of the atoms mentioned

Explanation:

Hydrogen bonds are the most intense type of intermolecular force that occurs between permanent dipoles of molecules, where the positive pole is always hydrogen, and the negative pole can be fluorine (F), oxygen (O) or nitrogen (N), since these elements are quite electronegative.

In other words, these atoms most strongly attract double-bonded electrons and are negatively charged.

An example to consider is hydrogen fluoride (HF) molecules, a polar molecule. In the solid state, its molecules attract, so that the positive (hydrogen) pole of one molecule attracts the negative (fluorine) pole of another.