If it's the one I'm thinking of, you should find the ship in the Racet system, four jumps from and to the Southeast of Igadzra. I thought that mission told you to find a named ship though, something like A.S.S. Lazerfunk, but it may not - it has been a really long time! And that mission stumped me on three different occasions before I ever got further in the Igadzra storyline - in fact I believe I found out where it was from the board here :).
Either way, yes it will be a specific ship that needs killing, which is why I think it will be specially named. And as for your other query, you cannot capture ships that you need to kill for missions - you will be told "You cannot board this ship." In some cases this is useful as you can tell if you needed that one when you're trying to find the last one or two of a big fleet; in other cases, it's just a safeguard against what you were alluding to. So no, capturing ? killing. In fighting big fleets, you can capture the fighters, but never the capital ships for missions.
Anyway, an interesting bit of info earlier in this thread - I never knew escorts increased your hyper-time! Does that hold true for captured as well as bought ones? The former were the only types I ever played with.