The tachinid fly (Trichopoda pennipes) is the #1 squash bug predator, but is not commercially available.
The pink-spotted lady beetle (Coleomegilla maculata) is an option that is widely distributed throughout the mid-Atlantic, southeastern, midwestern, New England and southern Ontario regions of the United States and North America. Be aware that it’s diet also includes pollen.
You can try planting flowers, such as marigolds, calendula, sunflower, daisy, alyssum, or dill nearby to attract beneficial insects such as tachinid flies.
To minimize squash bug damage:
Cleanup the garden at the end of the season to make it tough for them to survive the winter. Clean up all debris and leaves. Infected plants should be placed in plastic bags in the sun until decomposed or removed from site. Leave the ground bare of mulch for several weeks of cold weather.
In spring, protect young plants with floating row covers. These row covers keep the squash bugs away from plants but still allow air, light, and moisture in. Row covers must be removed later in the season to allow pollination.
Check the plants early for infestation. This may allow you to remove adults and eggs by hand. Drop the adults into a bucket of soapy water. Squash bugs lay yellowish brown to reddish eggs on the undersides of leaves in evenly spaced groups. You can kill the squasheggs by gently rubbing them back and forth with your fingers.
Early action also makes it easier to control the infestation with insecticides. Each adult that is eliminated can reduce the future population by several hundred. A first application should be made when eggs are first detected. Follow this with a second application a week or two later.
A number of insecticides effectively kill squash bugs. Look for products containing carbaryl or permethrin. (Looks like you've already found something that works!)
Insecticidal soap and homemade sprays work. Insecticidal soap is effective against adult squash bugs, but not against eggs, so you must still crush those by hand. Also, household soap combined with mineral oil and minced garlic is reported to be effective by the Henry Doubleday Research Association in England. Its formula is 3 ounces of chopped garlic, soaked in 2 teaspoons of mineral oil for 24 hours, added to a pint of water with 1⁄4 ounce of soap. Strain before using.
Rodale’s garlic spray is one bulb of garlic, one small onion, 1 teaspoon of cayenne pepper powder mixed with a quart of water. After soaking for an hour, add a tablespoon of soap and mix well. These homemade sprays are described in Rodale’s Chemical-Free Yard & Garden, edited by Anna Carr and Fern Marshall Bradley, Rodale Press, 1995.