Are you sure the eggs were black? Could they have been dark brown? This page has a picture of squash bug eggs, most of the way down the page. They come in a range of colors, from yellowish to dark brown. If that's what they look like, scrape them off and squish them. Here's another page with a pic of squash bug eggs

I don't know of a black and white bug. The worst squash pests I know of are the squash vine borer (a red wasp-like moth that lays single tiny eggs, which turn into larvae that bore into the stem of the squash and kill them -- though I think butternuts are pretty resistant to SVBs, as they are known 'round here). Then there are squash bugs. Then there are striped cucumber beetles, which are about 1/4 inch long, colored mostly yellow and black, with stripes along the wings (though they also come in a spotted form). SCBs can spread Bacterial Wilt, which makes plants wilt and die.