Bugs are a group of insects, characterised by their mouth parts. The group does not include ladybirds, despite their commonly used name of ladybugs. See this page. The word bug seems to be used of many insects and similar creatures for which it is a not strictly correct, but nevertheless convenient, term. This usage seems more prevalent in the States, but is gaining ground in Britain too.

Returning to lullabies, how about

Rock-a-by baby, your milk's in a tin,
Mummy has got you a nice sitter in,
Rock-a-by baby, don't get a twinge,
Mummy and Daddy are out on the binge.

Not exactly hostile, but a fair portrait of modern life.