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I think there is a possibility of other electronegative atoms forming hydrogen bonds, but N/O/F are the specific elements for which the hydrogen bonds are favorable and the strongest, so N/O/F hydrogen bonds are the only ones we should be concerned with.
The electronegative atom must have one or more unshared electron pairs (which oxygen and nitrogen have). The atoms must also have a partial negative charge. The hydrogen (has a partial positive charge) is attracted to the partial negative charge of oxygen or nitrogen which has excess electrons to share. I believe another reason is that there is a great difference in electronegativity between H and N, O, or F atoms.
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