It is possible to cast any spell as a counterspell. By doing so, you are using the spell's energy to disrupt the casting of the same spell by another character. Counterspelling works even if one spell is divine and the other arcane.
So they are the same spell. Spell Specialization should be fine. It requires Spell Focus, however.
Choose a school of magic. Any spells you cast of that school are more difficult to resist...
Special: You can gain this feat multiple times. Its effects do not stack. Each time you take the feat, it applies to a new school of magic.
Is there any reason to think that divine necromancy and arcane necromancy are different schools?
Well, the feat does treat the class you gained the spell from as relevant for at least some purposes.
Every time you gain an even level in the spellcasting class you chose your spell from, you can choose a new spell to replace the spell selected with this feat, and that spell becomes your specialized spell.
At best it's ambiguous. My instinct is to think that spell class matters, but I could see it going either way.
In the grand scheme of things though, I don't know that it really matters or creates a balance issue if you let it work with multiclassed casters. I could be overlooking something, though.
While "balance issue" is a matter of opinion, there is a consequence.
Personally, I don't really mind, since the PC Gark plans will have to be pretty gimped right up until 11th level (and, incidentally, she's planned for Age of Worms, so that could hurt quite a bit), but I am wondering if there's a clear "no" intended.
If there's no firm consensus, I'll probably let it slide. The difference is basically in requiring an extra expenditure of the feat, so doing otherwise would be little more than an inconvenience.