there is very little evidence to support the claim that Grey Squirrels are responsible for the egg thieving they are blamed for. Actually Corvids are far more common nest raiders.
I'm not saying it's not a worthy cause but may I also point out that the red squirrel is also a rodent and as such just as inclined to rob nests as the grey squirrel?
being ground nesting birds woodcock are also under threat from Hedgehogs, Badgers, Foxes, Stoats, Weasels etc etc.
I wish you luck in eradicating the Grey squirrel I truly do but it's impact on woodcock numbers will be minimal.