An eggshell is the outer covering of a hard-shelled egg
Egg (biology)
An egg is an organic vessel in which an embryo first begins to develop. In most birds, reptiles, insects, molluscs, fish, and monotremes, an egg is the zygote, resulting from fertilization of the ovum, which is expelled from the body and permitted to develop outside the body until the developing...

 and of some forms of eggs with soft outer coats.
Insects are a class of living creatures within the arthropods that have a chitinous exoskeleton, a three-part body , three pairs of jointed legs, compound eyes, and two antennae...

s and other arthropod
An arthropod is an invertebrate animal having an exoskeleton , a segmented body, and jointed appendages. Arthropods are members of the phylum Arthropoda , and include the insects, arachnids, crustaceans, and others...

s lay a variety of styles and shapes of eggs. Some have gelatinous or skin-like coverings, others have hard eggshells. Softer shells are mostly protein. It may be fibrous or quite liquid. Some arthropod eggs do not actually have shells, rather, their outer covering is actually the outermost embryonic membrane, the choroid, which serves to protect inner layers.