Most likely whiteflies...
See the link. Looks liek there suggestion is Neem-based pesticides(inorganic), but promoting predators such as beetles will take care of them(Organic).

Here is a link that might be useful: Infonet-Biovision

Thanks for the info, but that's not the problem--the eggs were clustered exactly the same as squash bug eggs, only they were WHITE--same triangular pattern and everything --maybe they were just "freshly" laid?

Could be on to something there.

Please post a picture

I read that squash bug eggs are white when first laid and then turn yellow/bronze.

Yeah, they start out almost clear on my plants, and progress through yellow to bronze-ish. Squish 'em!

Don't worry, when it comes to matters of squash bugs--it's squish first ,ask questions later--- but maybe it was an alien strain of ALBINO SBs--oh. well,---squish first...

For anyone not knowing about WHITE eggs on squash or cukes--well I found out that they are (by now) the infamous brown marmorated stink bug! I have dispatched quite a few by knocking them into a bowl of water and dish detergent--knocks them right out---they congregate between leaves and will try to avoid control by just falling off the leaf or stem--(what a strategy) look on the soil (they blend in)--and spray there-- I grab them with a plastic fork into the soap/water--I think insecticidal soap will work, BUT BE CAREFUL when you use it so it doesn't burn plants--best to direct spray the bug --maybe DE would work, too--I'm going to try it

During my morning search on my squash plant. There were round white eggs along with hatched brown looking bugs. I agree with Jacewell and they are the infamous marmorated stink bug. Photo included.

Congratulations, gratebrittian! Hope it becomes a life long hobby as it is for many of us!!