Let me just begin by saying I despise squash bugs. I don't remember having any trouble with them in our garden when I was growing up in Colorado, but they have been a constant nuisance since I've been gardening in Utah.
The first year sweet husband and I planted a good sized garden, we found these very cool little eggs on the bottom of our squash leaves.
They are usually laid in kind of a pattern and most times follow the veins of the squash leaf. Interesting. So, being the curious people we were, we thought we'd let them hatch and see what came out. Dumb. Don't do this. If you find some strange eggs in your garden, go look it up on the internet (which, by the way, wasn't nearly so full of information then) and maybe you'll end up here reading about squash bugs! DO NOT just let them hatch--get a sharp stick or rock and smash them! The eggs range from cream colored to very dark brown. The darker they are, the closer they are to hatching. Most times they are laid on the underside of a leaf, but some times they are on the stems as well. Just smash them all. It will probably destroy a little piece of your leaf, but don't worry about it. Sometimes I've found a leaf that is particularly heavily infested. I'll just take that leaf off the plant and destroy the whole thing. Most squash plants have plenty of leaves--they won't miss that one.
If you do let the little eggs hatch, or if you've been slacking on your squash bug squashing, you'll find these super creepy little grey bugs with black legs crawling around.
Don't make them your pets, smash them. You could also use some type of insecticide on them that works on squash bugs. I haven't found any that isn't just as much hassle as squashing the stinky little bugs since it has to be sprayed right on them and not just in their general direction. I really don't like using bug spray either, so we generally just stick to squashing them. Maybe you know something that works, I'm open to suggestions. These little guys will kill your squash plant. Yes, we had them crawling all over our squash plants that year and the plants withered and died and we had no zucchini to leave on our neighbors' doorsteps. Something's just not right about that.
Now, the eggs and the little creepy guys all come from big squash bugs that look like this:
They are commonly found hooked together, making fertile eggs, but are also found wandering around your squash plant single. They like the leaves, the stems, and the squash and can also be found around the plant on the ground. You'll find them when you're looking under the leaves for eggs. Smash them. They stink when they get smashed. Maybe hire your kids to squash them. Make it your family night activity to squash the squash bugs and their eggs. We have. Yeah, we're cool parents like that. Whatever you do, don't let them live and propagate out of curiosity! No good comes of it.