Just Don't know

It sounds like you need to go out to the woods at least in the beginning. Once you get to know the neighbors better, you may decide that shooting them at the feeders won't cause any issues.

I know the feeling. In suburban UK you have to assume everyone is anti. I take squirrels that empty my bird feeders on a tree 28 yards from my window. But I use a sub-12 foot pound .177. I suppose I could now use the .22 Wildcat. Also sub-12 foot pounds - even quieter.
It would be illegal to use my higher energy guns in my garden.

With low energy, it HAS TO be a headshot on squirrels for humanity's sake, which adds to the challenge.
They do, after all, make a fine chilli.

But I make sure no neighbours are about.

Are your feeders out of sight of your neighbours?

Billy, if they're the same neighbors you've been having problems with, I'd say wait for the backyard slayings. But during season, I say have at it. And take them a bowl of squirrel dumplings when you're done.

I have tons of squirrels at home as well. If I have been having issues with squirrels destroying my property, I will thin them down during season. AT any time they become a pest, I will take out the problem squirrel or squirrels(Legal in Ohio). Otherwise, I like watching them at home and with a dozen or so hickory trees out back, they tend to have plenty to eat and leave the bird seed alone.

I get what you are saying about shooting them at home. Not a challenge and I don't consider that I am really hunting them. Feels like shooting fish in a barrel. I always head to the woods for squirrel hunting. I like eating squirrel, so if my backyard was my only option to squirrel hunt then I would. As far as your new neighbors, if its legal than I would not change my ways for them.

Yeah, MarkUK, you need to post that recipe in the following thread!

http://airgunnation.dev/topic/recipes/

I don't know where you live, but in Oklahoma, we ate chicken and dumplings as a staple growing up. I haven't tried squirrel and dumplings yet, but I plan to fill my larder this fall/winter with squirrel so I'll have plenty to go around.

And who knows? Maybe it will change one day. Maybe the young husband will remember he's a man and will want to do man things.

If you consider them a pest you could get a DEC permit but you must bury or cremate them immediately. Would DEC consider a Weber grill as a crematorium?

http://www.dec.ny.gov/animals/81531.html#2

Chuckle...The Weber Grill a crematorium...
My neighbors are all good with the backyard shooting as I have done many favors for them thinning the squirrels that destroy flowers and garden plants. It took very little convincing as I had a large block party at my home and everyone there had a chance to shoot several of the quiet rifles on the target range. Free food, good conversation, a few fixed lawn mowers later and the fact that they participated in the fun of shooting sealed the deal.
Shooting the vermin at the shorter distances does not enter into the equation when they become destructive pests. I counted 24 squirrels in my Pecan trees and hanging on the bird feeders...time for a pellet order to Pyramid and fire up the Weber.