Surely, by the same logic, any atom with a moderate electronegativity (carbon, phosphorus, silicon) can form a polarise covalent bond with an atom with a high electronegativity (fluorine, oxygen). Giving that carbon, phosphorus or silicon a partially positive charge (just like hydrogen), allowing it then to form a 'hydrogen bond' with another atom with a high electronegativity (fluorine, oxygen). Is it called a hydrogen bond because it most commonly occurs with hydrogen? Or is it that a hydrogen bond with carbon, phosphorus or silicon is instead called a carbon bond, phosphorus bond and silicon bond? Since you don't need a hydrogen to make an atom high electronegativity partially negatively charge, surely those atoms around atom would be paritally postive to balance the charge?
Just looking at the electronegativities, there seems to be no reason why atoms such as boron, carbon or silicium — which all have similar electronegativities to hydrogen — should not undergo hydrogen-bond–like intermolecular interactions.
There is one key difference that separates hydrogen from all the other nonmetals that readily form 2-electron-2-centre (i.e. standard) bonds at ambient temperature and pressure: the lack of any core electrons.
Atoms like boron and carbon (or worse: silicon) have a low-lying 1s atomic orbital that is generally not considered to take part in bonding. However, its spherical structure surrounds the nucleus and effectively shields its positive charge from the outside world. Hydrogen, on the other hand, has a basically unshielded nucleus consisting only of a positively charged proton.
The positively charged hydrogen nucleus can now interact well with other orbitals on other atoms (typically part of other molecules). Any in-between core electrons will repulse the electrons that the electropositive element is trying to interact with. Hence, the stabilisation is greatest in hydrogen’s case.
In fact hydrogen bond is not a bond but a kind of dipole-dipole interaction so it is an inter-molecular force. Indeed these kind of interactions occur also with other atoms but not with the same intensity.
The hydrogen bond is due to the fact that there is a covalent bond between an hydrogen atom and an high electronegativity atom such as O, F or N. What happen is that because of the small size of the hydrogen atom the dipole is highly concentrated in a small volume, and this results in the high directional strength of the hydrogen bond. With bigger atoms the bond would not be so strong.
To be honest, I am unable to explain why hydrogen is in a special category when it comes to potential bonding, it just is. (The quantum mechanics would tell you that it does hydrogen-bond, and a thorough analysis of the QM might shed some light on it - if you want to go there. Personally, I've always been comfortable with just saying that nothing else comes close to the non-covalent (intermolecular) bonds it can participate in. Two other notes:1. Chemistry is "all about" electron density. Sure, it's quantum mechanical wave function probability distributions, but its all about electron density changes. Electronegativity is a convenient term to "invoke" to "explain" the effects of electron density on changes in bonding. This is very true. But, electronegativity is the "wrong" term. Technically, electronegativity is the property of the element in its neutral ground state. That is Co° and Co(+3) have the same electronegativity...even though we KNOW that the electron density is vastly different around them (both can participate in bonding with organics (organometallics)). So, the term is misused probably as often as it is not (if not more). Especially by beginning students (and their teachers!). I see no real harm in it - except when kids (or adults) go to the electronegativity tables and claim that say C in methane has a different electronegativity than C in ethane or formic acid. It's a slippery slope. 2. Not too long ago (10 years or so, couldn't (I hope!) have been 20), a paper was published showing that halogen bonding was a "thing". It remains to this day to be controversial. The Wikipedia article on h-bonding says it typically has bond strengths of 1-5 kcal/mole. The Wikipedia article on halogen-bonding says the bonds range between 5 and 180 kJ/mole. What's wrong with this picture?!! Well, 1 kcal ~ 4.2 kJ. So, 180 kJ is ~43 kcal. There's no f**king way halogen bonding is 8 times stronger than h-bonding. It's nonsense. X-bonding is so weak, that there's real debate about the usefulness of the concept (if it isn't significant then there would be no use in invoking it). My point is, separating out hydrogen-bonding for special consideration has proved itself to be a USEFUL thing. This is just not so with C, P, Si, O, or F (although the latter two often participate in h-bonding). It's just not very useful to consider say the Si in Me2C=Si interacting strongly with the O in Me2C=O While if you take a look at the difference in Electronegativity between them (using the tables) you'd not come to that conclusion, I think. Hence: not useful.