The idea that spiders lay eggs in or under human skin is a surprisingly common myth, but it's just that, a myth. All spiders are free-living hunters, not parasites. Spiders cannot lay eggs or live inside any part of the body. Spiders generally avoid humans as much as they can, but some species will bite if disturbed; a spider bite is a serious injury and should be treated as such.
It's possible that the origin of this myth comes from the life history of Dermatobia species, commonly known as botflies. The botfly will lay its eggs on or under the skin of a mammal, where the larva develops as a parasite, usually creating a swelling or lesion. These lesions can be quite painful and carry the risk of secondary bacterial infection as well. There are a number of species of botfly which utilize different hosts, including one species, D. hominis, which lays its eggs in human skin.