Question
I live in Austin, TX and have planted some squash
(specifically Pumpkin, Butternut, and Delicata). They grow
really quite well...for a bit. And then seem to start dying
off. The dying seems to start at the base of their stalk
and then slowly the rest of the plant dies.

Some images of the plants are here:
http://www.magosong.com/photos/plants/squash1.png
http://www.magosong.com/photos/plants/squash2.png
http://www.magosong.com/photos/plants/squash3.png
http://www.magosong.com/photos/plants/squash4.png

Random Info I have that may help:
I haven't noticed any pests in common with all three. The
butternut had a few aphids, and the pumpkin had a cluster
of half-inch red insects (not spider mites) on it.

I've noticed that around the dead area of some of the
plants appears to be an orange crumbly substance (visible
on squash3.png and squash4.png)

I have not treated the plants with anything except a recent
misting of water/dish soap/neem oil as an attempt to
address the problem.

They are potted in containers with organic local compost
and vegetable growing soil.

My water is slightly alkaline.

Answer
Bryan, it sounds like your plants have been infested by the squash borer. Squash borers are very common pests for all members of the squash family. These pests boar into the base of the vine, and eventually spreads a disease that causes the plant to slowly wilt near the base, and eventually die. Depending on how far it has traveled within the stem, it is not always easy to locate them. Unfortunately, once the borer does its damage, there is little chance for the plant to survive. The good news is that the fruit that has already set will continue to ripen and can be harvested.  

There are a couple of preventive measures you can take in the future. The first is to cover the plants with reemay while they are just seedlings. Reemay is a lightweight, spun-woven fabric that lets in air, water, and sunshine, but keeps out insects and pests. It is commonly sold at garden centers as floating row covers. Just be sure to remove the covering once the plant begins to flower to enable access by bees for pollination.    

The second option is to wrap your stem with aluminum foil. Many gardeners do this routinely with much success. I would personally recommend covering the plants when young, and then wrapping the stem once the plant begins blooming.

Attached is a website that will give you additional information on the squash borer.  

http://www.ag.ohio-state.edu/~ohioline/hyg-fact/2000/2153.html    

I hope this information helps. Please write again if I can ever be of assistance.

Regards,

Mike