My guiding maxim would always be "What would any normally compassionate, humane individual do with a sentient creature?" but clearly other people are not guided by such considerations.
The problem with this approach Shaker is that it only works in an unnatural vacuum. As has been said, grey squirrels are vermin, but more than that they are an invasive species - such as mink, coypu (used to be), Japanese Knotweed, etc. - and have a deletorious impact on the UK's native species. Furthermore it damages forestry and other non-sentient organisms thus creating problems for that side of things as well. As far as I'm aware, there is no natural predator for grey squirrels here in the UK (though a dog my daughter used to foster used to make a good fist at doing so!!)
Are you suggesting that it should be left to roam unchallenged, and that its 'sentient being' status overrides its potential for damaging native species and their habitat?