Hydrogen bonding may be understood in different ways. My own preference is it usis a form of delocalized covalent bonding. This is discussed in this answer.
This molecular orbital description does not require the bonding to be polar, although the polarity commonly associated with hydrogen bonds (hydrogen primarily attached to a much more electronegative atom in both the primary bond and the hydrogen bond) favors more bond delocalization and thus a stronger hydrogen bond. Answers to this question explore the possibilities, including one where that cites hydrogen bonding to carbon atoms.