Here's some non-poisonous suggestions for ways to work on dealing with your problem, though with an infestation of that magnitude, it may take a couple years:
Squash bugs seem to prefer yellow squashes, so it helps to grow zucchini, cucumbers, melons. If planting winter squash, try planting varieties that have shown resistance to squash bugs. These include Butternut and Royal Acorn. You can try planting things that are supposed to repel the bugs in with your squash. This includes garlic, onion, tansy, radishes, nasturtiums, marigolds, bee balm and mint. Planting your squash later in the season, once the majority of the squash bugs have already hatched and perished can help. Try putting boards down on the ground near squash plants over night. In the AM you can get rid of the bugs that have gathered under the boards. Dust around the plants with diatomaceous earth. Mulch heavily and then cover your plants with floating row cover. tightly secured. I spray plants with an organic repellant made like very hot salad dressing with salad oil (to help it stick to the plants), a little vinegar, lots of garlic, black pepper, hot pepper, some tomatoes, aromatic herbs, mint. Blend it up in water, strain, and spray on plants. Not fool proof, but it definitely seems to help.