White caterpillars would not be the Eastern Tentworm Caterpillar and it is too early for their cousins the Fall Webworm which is not white either. I'm not familiar with a white caterpillar that spins a web, yet. As a rule none of the webworms really do much harm to the trees and shrubs they inhabit, but the webs are unsightly, and if a tree is defoliated (not usual) over a number of years they can weaken that tree.
If control is desired spraying the leaves with Bacuillus thuringiensis - Kurstaki is the prefered method, you can cut the section of branch these buggers are on off the shrub which may disfigure it. Never, ever burn the web out since that can cause more damage to the shrub then the caterpillars would. Any leaves they do eat will grow back in later.

Here is a link that might be useful: About Eastern Tentworm Caterpillars

Might be sawflies -- If so, they eat leaves, usually in groups.

any chance of pictures?

There are other caterpillars that make tents, besides the two you mentioned. However, depending upon where you are located, fall webworms would certainly be out and about at this time. Some color strains are have very white hairs. All fall webworms species are very hairy.

As Jean mentions, a picture would solve the question of identity right away, as long as we could see something of the nest, its location on the plant, and maybe even of the critter.

Thanks. It was raining but we had a break so I went out and took some pictures.

*My thought was to prune off the branch and possibly burn it in a firepit since I was planning to cut back that branch anyway. I would rather not have to spray with any insecticides unless I have to. But still, I want to know exactly what caterpillars I have for next time.

Fall webworm.

Thanks. That's what I thought. I read that although they are called "fall webworms", they can start as early as June in some areas of the country.

Thanks for taking the time to provide good pictures.

Your best control will be to simply clip off the offending nest and dispose of it, caterpillars and all. Unfortunately, there may be more egg masses on your shrub. Look on the underside of the leaves for drifts of many little eggs.