It is a small photo of the bug...but I think it probably is.

That's a spotted cucumber beetle. There's another similar type with stripes.

Do they damage zucchini plants?

Yeah, they do most damage to cucumbers, but they are a minor pest to zucchini and all the cucurbit family of plants. The worst problem with them, in my estimation (perhaps others can offer more information,) is that they carry, and spread disease.

My cucumbers are doing great and the zucchini right next to it has been dying. I thought maybe this was the culprit.

It could be, but this time of year the most common cause for zucchini dying is SVB (Squash Vine Borer) damage. Look at the stem of the zucchini. Start at the base of the plant where the stem comes out of the ground and look for little piles of 'frass'. You can search for images of what the frass and SVB damage looks like.

However, if the cuke beetles are bad, they certainly could be aiding, or even causing the demise of the zucchini. The other insect common to zucchini is Squash Bugs. The nymphs especially will suck the life out of zucchini. Do a search for what they look like. The nymphs are usually a whitish color, and can be found on the underside of leaves, and all around the base of the plant - often hiding (like the adults) under leaves that are touching the ground.