No, spiders don’t lay eggs in human skin. It’s a myth.
Other insects may do this, like the mite; that create the skin disorder scabies, but spiders are far too concerned with the well-being of their offspring to entrust them to the likes of us.
Warmth isn’t what mother spiders are necessarily seeking for their egg sacs, anyway, security is.
And that means protection from predators like birds. Yes, it’s true, some birds eat spiders.