Those aren't eggs, but baby aphids. Adults bear live young...each of them female who will soon be having their own babies.

What ... there are no male aphids?

Correct. Mama ladybugs don't need males during the growing season. The ladies do it all on their own.

It's called parthenogenesis. Only at the end of the summer /fall are some males born. They develop wings and flit around to mate. Once mated, a female will lay eggs....which is how aphids overwinter in cool climates.

Interesting-and frightening-to know that each aphid is actually born pregnant!!

And each adult can bear 50, 75 or more young in her life....each of which can begin their own reproductive stage in little more than a week! In warm climates, Aphids can remain active all year round.

Thank goodness for the many natural controls out there, right?

I stopped using pesticideds of any kind in my yard about 9 years ago and so glad I did . As rhizo 1 pointed out there are many natural controls. This year the aphids got really bad , they blaketed my new crop of rosebuds . one morning i went out to do what I could by wiping them off (this takes about an hour ,I had a lot of rose buds) and low and behold there was a flock of 20+ tiny happy churping birds not much bigger than a humming bird;feasting on the aphids. A half an hour later they were off and the aphids were totally gone!. and the lady bugs are doing a fine job keeping the aphids under contol. Its so neat the way nature works.

I have a spirea that get aphids every spring I leave them alone because I know the ladybugs will come. I especially love seeing the ladybug larvae and I have fun moving them to spots that are really needy.

Mo, you might have seen actual hummingbirds.....they love sugary aphids.

I grow milkweed just for the purpose of growing MW specific aphids (along c Monarch butterflies). These orange aphids attack nothing else and get so thick on the MW that sooner or later the Calvary comes in and cleans house. Ladybugs, Lacewings, Hoverfly larvae and parasitic wasps all then move thru the garden mopping up as they go. Sort of like a vaccination.

Alaways a delight to find others are as facinated by nature and how it works. I move my lady bug larvea too. The other day I found three lady bugs each was in a different stage of growth(I believe they have five). I thought this was so neat I took each one on its own leaf into the house to show my husband who tried but wasn't nearly as excited by my findings.
Im not sure what kind of birds they are (I'll have to post a pic or do a little more research) but the hummers at my house are very territorial and these little guys work as a team or flock.