I live near Formby Red Squirrel reserve and a member.
They have certainly caused a large yes but probably not for the reasons most people think. It's a number of reasons. They actually live alongside the Reds quite happily and that's why the Reds suffer.
Greys are immune to Squirrel pox virus which is the biggest killer (80% of Reds by 2008) and the grey get it but act only as carriers and pass it onto Reds which unfortunately have close to 100% mortality rate. Grey live in much higher density than Reds so overtake their feeding grounds and living spaces, plus if there are any diseases they pass them straight on and with a smaller population, proportionally more die. Greys are much more efficient at feeding and storing in woodlands, and being bigger also store more fat in their bodies so tend to survive for more winters.
One of the reasons the reds are doing so well on this reserve is because they feed on the seeds found in conifer pine cones. Reds can get to them but the Greys are too heavy so just move on elsewhere and don't bother living near them if there isn't much of another food source. Plus, the volunteers locals "remove" any Greys that are sighted in the area.