I'm so sorry about your beautiful squash plants. Having battled these bugs myself, I sympathize.

There's no easy, one-and-done cure for squash bugs (at least, not if you want to be able to eat the squash afterward), but there are, nonetheless, effective ways to keep their numbers low enough to allow plants to survive and produce.

It all begins with monitoring. Once your plants are in the ground, check on them daily to inspect for squash bug adults, nymphs, or eggs. Be sure to look under the leaves and on stems, where they like to hide. And don't forget to check all your cucurbits (squash, melons, cucumbers, etc.), as any of those can be a host plant for them.

You can also protect your plants with a light row cover until they reach flowering stage. Just make sure it doesn't shade the plants too strongly, weigh them down, or trap excessive heat. And you'll still want to check on them, in case any bugs found a way in.

Once you've spotted a squash bug, don't wait to attack. Treat according to the life stage—

For adults, the simplest thing to do is pick them off by hand and, as you put it, extinguish them. This can be accomplished by squashing them (no pun intended) by hand, stepping on them, or throwing them into a cup of soapy water. (I prefer that last option, since it avoids them emitting their pungent odor.)

Nymphs are the most susceptible to pesticides, but even then the effectiveness is limited since they often require contact, and squash bugs are good at hiding, especially on the undersides of leaves, where treatment might not reach. Instead, mix up a solution of water and insecticidal soap in a spray bottle (per the label), and directly spray the nymphs, which are usually clustered together near their hatching site for a while. You can take care of a lot of them at once that way and put a nice dent in their population.

Note: While you can use a dish soap/detergent in lieu of insecticidal soap and many sites recommend this, I would suggest you exercise caution in doing so. Evidence suggests most are too harsh and can damage plants. If you do choose to use dish soap, though, make sure to use one that is relatively gentle and biodegradable—just look for the EPA Safer Choice seal on the container. (You can also view and search a list of all products that have this label on this EPA page.)

Eggs (and adults, for that matter) are also susceptible to the insecticidal soap, but you can also just manually remove and collect them (you don't want them falling to another leaf or the soil) or squish them where they are. One handy trick I see used a lot is to wipe them off with tape—usually masking/painter's tape or duct tape, but packaging tape also works. The idea is to choose a tape that has strong enough adhesive to cling to the eggs without tearing the leaf, so you might have to experiment to find the right one for your plants. After you collect the eggs, you can fold the tape in half to contain them (or again, squash them!).

There is one unconventional tip you can try if you have space enough: Research has given us strong evidence suggesting that squash bugs have species preferences and will favor blue hubbard squash above all other kinds of cucurbits. Knowing this, you can plant hubbard squash as a trap crop, which will draw them away from your other cucurbits. This would also help limit their dispersal, making it easier for you to stay on top of their control without having to do as much hunting. Of course, if you don't plan on eating the hubbard squash, you can also take it a step further and treat it with a systemic insecticide, which would mean any squash bugs that try to munch on those plants would automatically die, saving you a lot of hassle. But use with caution: Systemic pesticides do go to all parts of the plant, including pollen, so in addition to not consuming the produce from these, I would suggest plucking off any blossoms before they bloom, in order to protect your bees.

There isn't a lot in the way of biological control because squash bugs apparently taste as bad as they smell, but the most of the generalist predators—spiders, praying mantises, etc.—aren't as picky and will make a meal of them. Chickens don't like them (I speak from experience here), but guinea fowl reportedly do.

There are pesticide options, but their effectiveness is limited, since they require contact and squash bugs are good at sheltering under leaves and such. Furthermore, some are only effective against nymphs, and ones that are effective against adults are generally toxic enough to unintentionally hurt other insects as well, such as bees. You can find lists for all your chemical options in the resources linked below. Always read and follow all directions on the label.

During the season, clean up any leaf litter the squash bugs might hide under. When the growing season is over, dispose of any plant material that might harbor them. You can compost it (and I'm all for that!), but I would recommend burying it a bit in the pile; you want to prevent any live ones escaping before frost comes and make sure they get thoroughly cooked.

I sincerely hope this helps. I know it sounds like a lot of work, but I have found that if I make a concerted effort early on, I will have far less work later.

Best of luck to you next year!

PS: While you're bug-hunting anyway, you might consider watching for the elusive squash bee for fun! They weren't spotted in Oregon until 2016, and now OSU Extension is doing a Squash Bee Survey, asking growers to send them pictures of any bees they see inside squash flowers in the morning. (No need for you to identify them.) You can read more about that cool project here.

Resources

https://pnwhandbooks.org/insect/vegetable/vegetable-pests/hosts-pests/cucumber-squash-bug

https://ipm.ucanr.edu/home-and-landscape/squash-bugs/#

https://ipm.ucanr.edu/agriculture/cucurbits/squash-bug/#gsc.tab=0

https://extension.usu.edu/planthealth/news/squash-bug-trap-crops

https://extension.umn.edu/fruit-and-vegetable-insects/squash-bugs

https://www.pubs.ext.vt.edu/ENTO/ento-578/ento-578.html

https://extension.colostate.edu/resource/squash-bugs-in-the-garden/

https://extension.okstate.edu/fact-sheets/hla-6040-how-to-identify-and-managing-squash-bugs.html