Grey squirrels are accused of reducing the bird population. It is actually a quick issue to deal with, because an extensive, government-funded study has been conducted and concluded that it is not the case.
“Analyses of large-scale and extensive national monitoring data provides little underlying evidence for large-scale impacts of widespread avian predators and grey squirrels on avian prey population.” (Newson, et al., “Potential impact of grey squirrels (Sciurus Carolinensis) on woodland bird populations in England”, Journal of Ornithology, 2010)
It is true that squirrels, both grey and red ones, are opportunistic feeders, and, if they come across an unattended nest, they might take an egg. But the point is, they are not predators, they will not deliberately seek out eggs, they do not hunt the chicks, and what they do is statistically negligible.
Actually, the presence of grey squirrels can assist birds. When grey squirrels strip bark (which does not, by the way, happen very often – see the section on damage to forestry), it encourages fungus growth and insects – food for birds.