Those big tree roaches (Palmetto bugs for
you FL folks) have an egg case about 3/8" long, dark brown.

Aarghh!!! I'm almost afraid to ask, but how big is an adult tree roach?

I think the record is about 1 1/2" (some bug spray company has an annual
contest) but they certainly look a lot bigger when they are walking across the
ceiling over the bed.
It is technically called the American Cockroach. Take that you wimpy German and
Asian "also ran" bugs!
Just be happy we don't have those 3" hissing mo' fo's