Dawn said:

I'm not sure what I'm looking at. At the base of the plant there is
brown spongy slimy stuff where the leaves and runners branch off the
main plant. Some of the stalks look a little chewed up, but it doesn't
look like borers have hit it yet. Or maybe they have and I just don't
recognize it.

It sounds to me like borers -- has the weather been damp where you are (or,
are you watering frequently)? That would keep the frass wet and encourage
rot to set in.

Here are some images of vine borer damage on squash. Remember,
more sap and moisture might alter the appearance of the damage:

http://www.extension.umn.edu/yardand...s/squashvb.jpg
http://www.hort.purdue.edu/rhodcv/hort410/8021a.jpg
http://www.urbanext.uiuc.edu/bugrevi...ineborer02.jpg

I looked at several websites that talked about growing zucchini but
haven't seen reference to anything that looks like what I've got. It
almost seems like the plant is rotting. I'd like to save it if I can.

You can try fishing the borer(s) out with a thin, flexible wire with the working end
hooked over a little. Or you can inject the stem above the damage with Bt solution
or beneficial nematodes.

The 'traditional' way of salvaging squash is to slit the vine to get out the borer,
then bury it. This is more practical for vining squash rather than zucchini, as
they will root in many places along the vine as it runs.

--
Pat in Plymouth MI ('someplace.net' is comcast)

Any technology distinguishable from magic is insufficiently advanced.
(attributed to Don Marti)