I really don't understand the whole aggressive and stand-offish attitude people have on here. Part of it is the internet, and part ignorance I guess.
Fact is, they're neighbours and not going anywhere. At best the dog is just being a dog, so shooting it punishes it for the neighbour's inability to figure things out. The neighbour herself came over to apologise when she understood what was going on, and furthermore tried to help solve the problem the only way she could think of. That shows she's reasonable and open to your own solutions, as she feels bad and is trying to make things right.
I'd say that you explain that there are snares and traps around that her dog would likely find if it's finding bait piles, that there are coyotes that are sometimes rabid, and that lots of land owners don't care about shooting any animals they see as trespassing on their land. Tell her you like her dog and you're telling her because you don't want to see it get hurt and didn't think she was aware of the situation. Then give her some game meat, or invite them for dinner and cook some up for them, if they don't know how to cook it. Odds are they'll keep their dog on a line and be a long way towards being converted from Citiots to newly appreciating rural owners... rather than making enemies and ending up grumpy and waking up to their dog taking a dump on your front porch.