And it just figures that we're 3,000 miles apart! If I were closer, I would offer in-person help as I know you are troubled and frustrated. And I would indeed offer my property, were we closer (as a wildlife biologist by training, I am aware of the technical issues with potentially crowded territories as I write this...)
My gray squirrels and chickens have a very peaceful co-existence (if anything, the squirrels are a bit afraid of the chickens!). It sounds like your situation is different and since you have itty bitty ones, which I didn't previously realize, they may be vulnerable to critters that can edge their way in small spaces (even if not squirrels, then rats, snakes, whatever you've got there). The little munchkins are vulnerable to so much and I too would be worried if squirrel-sized critters can get in - I would be feeling very protective too. If a squirrel can get in, then so can some even more serious threats. And I agree about Shake Away as I tried that once for some rats that had discovered a coop at another location. It didn't work - the only think that did work was making the coop rat proof with hardware cloth. I know there are a lot of other squirrel repellents but I don't know their efficacy as I've never used them. There seem to be really great reviews on some.
Drowning is a pretty awful way to die - some would say if you have to kill to do it fast with a gun - of course, accuracy is crucial in that case. I don't have a gun myself - though some days I wish I did! (symbolically anyway, for the dreadfully irresponsible people letting their dogs run loose).
I am not sure why your squirrels are so interested in coming in when this time of year holds so much else for them elsewhere - mine have all but disappeared, exploring and eating what spring is offering them, showing no interest right now in chicken food or chicken housing. I wish yours would do the same. I do appreciate your dilemma and wish I could help.
JJ