For your initial question, if your neighbour's dog gets onto your place and attacks your stock, it's no good saying 'but it's their responsibility'. It doesn't really matter whose is the legal responsibility, the bottom line is, it's your hen that's died, so you should have made sure the dog couldn't get in. In addition, they should have made sure their dog couldn't get out....
I know in this case you have settled it amicably with your neighbour - which is great, and so unusual! We settled a similar problem with our nice neighbour when his cows flipped up their heels and jumped into our field to demolish a load of newly planted hedging plants. We went halfers on costs to add extra height to the fence so the cows, and the steers which are often in there, can no longer jump over. We would never have been able to come to such an agreement with the nasty neighbour on the other side, so we just make sure that fence is now secure ourselves. Who cares really if he should have done it? It would be our livestock which would suffer if we didn't do it.
I hope your little surviving hen is ok. They can die of delayed shock unfortunately. I'm reading everyone's suggestions on this as we had an attacked hen (by grandchild
) which did die several days later, even though she had only been chased and not hit, as far as we could see.