I'm such a noob at this, please educate me if you're willing. After two weeks, I've learned that it's not aphids in my lanai garden. These little beige and brown eggs all over the gardenia and cucumber leaves are WHITEFLIES. I know this because they flock and scurry away when I blast the plants with my spray bottle. Right now, they're very low in numbers, but eggs were all over my newer cucumber plant leaves. Hundreds.
So I did some checking online and apparently, whiteflies are a problem everywhere. There are all kinds of toxic and homemade remedies. One is to use a plastic yellow cup, fill it with motor oil. Apparently, the adult whiteflies can't resist the color yellow. That would explain how they like the cucumber plants — lots of bright yellow flowers. And the gardenia leaves they like best start out very bright yellowish-green.
One person wrote that the eggs are on the soil, too. Maybe my squirting the eggs off with water on the gardenias isn't enough.
I have a bottle of neem oil treatment that was for the gardenias, but I also used it this morning on the cucumber plants. I hope that works. The newer plants have been decimated. They were so vibrant and lush, grew so quickly. Now it's a jungle of dead brown leaves. I have hope, however. Since I've been paying more attention to the older cucumber plants, squirting off the whiteflies often, it's made a comeback. In fact, it's been sprouting new female flowers, mostly too small to pollinate. But this morning I discovered a HUGE new female flower with the fruit already 2 inches long. I haven't seen one that big since the flower that turned into my only cucumber harvest so far. That was about a month ago.
So there are other ways to battle the whiteflies. This thread at GardenWeb helps. I'm hoping for great results. Man, I hate these whiteflies.
Update: More great educational info on killing whiteflies at The Hot Pepper.