A core part of the scientific method is that you attempt to isolate a single variable at a time. If anything, all this suggests is that this was a better diet-controlled sample population for measuring the correlation of eggs and Alzheimer's than the general public. That said, the methodology of this study does not allow for inferring a relationship between Alzheimer's and meat consumption in either direction.
It is very difficult to do that in biological systems where doing A in isolation can have the opposite effect of doing A while also doing B.
There could be an interaction with the diet though. For example, what if the nutrient in eggs that prevents Alzheimer's is something that also occurs in meat?
Also, it seems likely that among this population many of those who don't or rarely consume eggs are vegan or almost vegan, so it might be more accurate to say that veganism is correlated with Alzheimer's.