Dill is a host plant for Black Swallowtails, but usually the eggs are white/cream colored.

Are the eggs clustered together?
I recently found this cluster
http://www.butterflygardening.org/graphics/Eggs%20on%20Dill%20-%20June11th.jpg
They turned out to be an unknown very small insect - approximately 1/16: long.

Aphids will eat larvae and eggs.
Here's a short article I wrote about the topic.
http://www.butterflygardening.org/#13

Try Bronze Fennel it's a hardy perennial and you only need to buy it once.

Here is a link that might be useful: Aphids, Eggs and Larvae.

Aphids are NOT equipped to eat insect eggs or larvae, nor are they interested. However, ladybug adults and larvae are. I hope that your article (which I could not open) didn't confuse that issue. Ladybugs and their larvae are quite opportunistic predators, and will take advantage of a wide variety of food sources.

That cluster of spindle shaped eggs looks like those of a ladybug, but could be that of a couple of other beetles, as well.

I went out and took a picture today of the eggs, which are not clustered together, they are each individually attached to a separate plant. I also took a pic of the fennel I purchased with two cats on it. I noticed a lady beetle larvae in my patch of dill and wondered how safe my eggs and cats are with this around. Now I know. But, how do I safely move the eggs. If I pick the dill, it will wilt. I don't have a cat house yet, though I have all the materials and will make it this weekend. What should I do? How can I keep them safe without harming them?

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How do I keep these guys safe from lady bugs until I get a cat house? (see pics above)

They'll likely be OK .... When they get big, they look like steak dinners for wasps...but I've raised many a BST outdoors.

Now....if you camaflogue the plant with a bunch of other plants, harder for wasps to find, and easier for cat to molt.... perhaps you need to get a screened in cat house sooner than you think?