Hi tchamber!
I had the same issue this year! This is my first ever garden so I over planted in a very big way. I mean lets face it, it's hard to image
seeds THAT tiny will turn into such large plants! Pretty cool tho! I had to thin my zuchini, cucumbers, and even my tomato plants. Once the plants were big enough to transplant I moved some of them to another area in my garden, and even gave some of them away to my Mom who decided to use them in a container garden. Unfortunately some of the plants had to just be pulled out to make room for the others. But I'm glad I bit the bullet now because the ones I left are much happier and healthier.
Also, I can't tell if you said the plants are
growing up the chicken wire or not, but have you thought about using just plain old tomato cages? It's def not a conventional use for tomato cages, but they come in varying sizes for ooober cheap and if you place them around zuchini or cucumber plants they automatically just climb right up! Saves on space but also alows for air flow like Marcia mentioned! One small tomato cage will support 2-3 zuchini vines.
Good luck and let us know what you come up with!