You may have the insect most hated by growers of squash - the squash bug. These bugs suck the juice out of the plant and the adults are nearly impossible to kill, at least with anything chemical free. To be sure this is what is attacking your squash, look on the underside of all the leaves. If you see clusters of copper colored eggs that are difficult to scrape off, you have squash bugs. First, kill any adults you see. The best remedy I've found is making a mixture of beer, water, Tabasco and a drop of dish soap. I've also read online that just pouring beer around the plants works so I've been doing both this year. Apparently we didn't get all the adults because a cluster of eggs keeps showing up on the same leaf but at least they are starting under control. This spray will kill any nymphs there may be but you will still need to scrape off any eggs you find. They do not come off easily. Make sure to kill the eggs.
Hope this helps. Good luck!