Earlier quoted context omitted.
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.
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.
2. They modeled swapping eggs for nuts/seeds/legumes and saw the same risk reduction. (Table 4)