I have actually heard of the alzheimers concern with anti-perspirants. The problem, as I have heard, is because anti-perspirants contain aluminum, which enters your body when you put it on. Now, I have also heard that aluminum dish-ware can cause alzheimers, especially when used for cooking. So, it is my assumption that this is true of the anti-perspirant ingrediant too. However, the aluminumn in anti-perspirant is different than that used in dish-ware, so this might just be a myth of sorts that was taken from one alzhiemer-aluminum association and applied to all things with the 'a'-word.
I have not heard anything about the breast cancer risk, though, but if breast cancer is in anyway caused by aluminum, then it could very-well be true.
As a side note, technically, deoderants do not have the aluminum, since they are not meant to prevent perspiration, just cover odors.
Jon