Wow, that's some freaky deaky stuff. I'm going to hazard a guess that it's a slime mold in its early stages. I reminds me of some slime molds I have seen in my lawn. It also looks like some pictures here:
Slime Mold Pictures
Although I've never seen one growing on a live plant. If it were my zuke plant, I personally wouldn't worry about eating any fruits from it, unless of course, it was covered by that organism, whatever it may be.

It's a slime mold. Not a problem for plant or man. If your plant died, look for a different cause. Wash off the squash & eat it.

Base of plant? Read up on vine borer beetle.

http://www.theprudentgardener.com/Zucchini.html

Here is a link that might be useful: Vine borer beetle!