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Those eggs look like squash bugs.. kill them with extreme prejudice (nuke 'em from orbit prejudice).

Regular dish soap will kill the squash bugs. You will have to scrape/smash off the eggs. Those little bugs are the babies.
They will ruin the squash.
mix 1/2 tbsp. of cheap dish(no Dawn) soap in a 1gl sprayer. Spray the bugs directly. It will take a few min. for them to die. You need to fine the mom and do the same thing. Try hitting the plants with the water hose(after killing the babies) she will climb on top of leaves to get away from the water. They don't like wet.

My nemesis! Duct tape works good to get those eggs off. I have also used a grill lighter torch and scorch them. Either way you need to get every one if you can. Check the underside of every leaf. What Campy says about the dish soap is true, I sometimes get so many I use a 1ý gallon pressure sprayer. I do use Dawn however. Campy should I not use Dawn?
If you are not sure that you get them all, in the fall you need to burn your vines etc. those bugs will winter over in your soil and come up again next year. I leave one undesirable pumpkin and vine late in the season to trap all the squash bugs and kill them.

Here is the pumpkin and vine I left at end of last season. Leave it there several weeks and go everyday and spray them with the soap water. Spray them until they quit coming or are all dead. Then I till the soil several times after it frosts. Tilling will help to bring any of them to the surface that have gone into the soil. Then hopefully the frost will kill them.

Sid23 I don't use Dawn anymore. It has a lot of that degreaser in it, more than the cheap $ store stuff. I once defoliated 20+ roses with dawn. The leaves all turned yellow and fell off. Roses did all survived. Agent orange comes to mind!