I've only had this happen when we lived in Mexico, and there it was a combination of soil ph and nematodes. Hopefully farmerdilla will show up. I bet he'd be able to give a more specific answer.

George
Tahlequah, OK

What do you mean by the rest of the plant seems fine? If the leaves are healthy and the plant is producing (which is what I would consider fine) what is the problem? Is the squash actually dying or do you just think it looks bad? To me it sounds like you are describing the base of a mature squash. As the plant grows the base turns brown and almost looks like a piece of wood. In the world of big pumpkins most people refer to the base of the plant as the stump because of the way it looks (and its size). Here is a picture (of an Atlanitc Giant) that I found, if this is similar to what you have then there is no problem at all: