As I understand it, the killing of grey squirrels is acceptable if done in a quick and humane manner. A lady I used to walk with, had a special cage into which the grey was enticed and quickly drowned in the fast flowing stream of her garden. Her garden was large and at one point she was overrun with them and they got into the roof void.
In the South Lakes, Cumbria, there is a telephone number to ring and someone will come round and humanely kill the squirrel on your behalf. This is not only in private places but also in local woods etc. They are so successful in this, that the reds are making a much wanted comeback.
There will be a debate about this is Parliament in the new year. I think mass culling is the way forward.
Same as you Henrietta, the squirrels have done a lot of damage in our garden and constantly scare off the birds.