No, probably not Rick. But eggs could be washed down to the base of the plant, into the leaf axils or onto the soil surface and if conditions are right, hatch and the little crawlers climb back onto the plant.
Any plants that come into my house get a dusting of DE on the soil surface, just as a precaution. Down here, it's very common for ants and roaches to hide/nest in house plants, warm moist environment etc. For that reason I rarely keep any plants in the house, although some of my orchids will be coming indoors tonight for a sleepover. (cold outside . .. we are having 'winter' for a night or two)
We also have a beastie here called a Sri Lanka Weevil, and I tried an experimental dusting of DE under a plant outside to see if it would slow them down. They feed on the leaves of plants, the adults mate on the underside of the leaves then the females drop down and burrow under the soil or mulch to lay eggs. They are singularly resistant to most insecticides, but I was thinking possibly the DE would help to break the life cycle as the new nymphs begin to climb out of the soil onto the plant again. Meanwhile a family of mockingbirds developed a taste for the adults, so the plant was saved in spite of my efforts. Didn't see any re-infestation but it may have been the DE or it may have been the birdies.
