From an ecological standpoint, they do indeed represent a problem. When species like the grey squirrel and red squirrel evolve within the same community, they will find a natural equilibrium to share resources. It's called niche specialization. Each species will focus on one particular niche to avoid competition for resources. This resource partitioning while reducing competetion for resources, also allows for the sharing of limited resources.
Now, when you add a grey squirrel to a region where it does not naturally occur, it will overcompete. The trophic structure can quickly tip the scales to favour the invasive species. Typically, invasive species flourish, because predators will continue to prey on their natural food sources, and the resources of the natural species will be reduced because of the invasive competitor. This results in a decrease in carrying capacity, and this extra pressure can cause species to dwindle low enough that they will not be able to recover.
Suppose for instance that juvenile red squirrels prefer the buds from a certain tree. If the grey squirrel reproduces more prolifically, and the young mature faster, they could overgraze that food item which is critical to the development of juvenile red squirrels.
Invasive species are nasty. One reason we should be very diligent when moving about the continent, or from one continent to another. It doesn't take much.
