I seem to be unable to get a clear answer from the book on the matter of if zealous hatred works with psychic bolts. From what I read in the section of the book that covers the attck actions, it appears that only attacks made with melee or ranged weapons count as such. Is it correct by RAW to say that psychic bolts do not get the benefit of zealous hatred?

I'd say it works. The section on Zealous hatred (p242 BC Core) itself says "When rolling damage after a successful attack, if any die rolled results in a natural 10...(rest of the description etc)"

A psychic attack is still, for all intent and purpose, using the Attack Subtype of action as defined on the details of the power so it would be eligible for a critical hit result.

I'd allow it as well.

To be more precise, I'd allow Zealous Hatred for Attack Actions (be it with a weapon or psychic power) that inflict Wound Damage, and only for the component of the attack that inflicts Wound Damage. That is to say, if, for example, you hit somebody with an attack effect that does both Wound damage and Characteristic damage, Zealous Hatred would only be triggerable by the Wound Damage, not the Characteristic damage.