Can spiders lay eggs in your skin?

No. Spiders go to a quiet place, make a sort of cup out of their silk, lay the eggs in the cup, and wrap it the rest of the way up.

Spiders do not even want to be on your skin. I just tested that once again today when I was cleaning house and found a little "second cousin" of a black widow. It was coming down a thread of silk, so I held my hand underneath it. The instant it touched my hand it knew that something was wrong and went right back up its line of silk to get away from this noxious landing place.