Hi Jonathan, 

Unfortunately this looks like the fungal bodies of Southern Blight and not any type of insect eggs. It thrives in the heat humidity and moist soils. There isn't a product that can control this. You can remove the foliage and try to big up the soil in that area. You may want to dig up the roots and surrounding area to try to get rid of most of the fungal spores. You most likely won't be able to get rid of all of them. They won't effect the roots of the hosta but the hosta will most likely get reinfected again next season when the weather heats up again the spores start growing in the heat. Alternatively there could be another fungus or bacteria that infects the Southern Blight and it won't damage the plants again next year. 

Typically the recommendation is to dig up the plant and the soil in the general area about 6 inches down and discard it and replant. You can replant with another hosta as Southern Blight infects most herbaceous material and there is no gaurantee it won't move into the area again. It doesn't however spread to your whole garden so just containing it in this area and trying to remove what you can is the best you can do. You can mix in a little compost in that area to help add beneficial soil microbes as that wouldn't hurt. 

Let us know if you have further questions. 

Emily