FET analog switch example of "The Art Of Electronics"
You are incorrectly understanding the phrase:

negative signals would cause the FET to turn on with the gate grounded

If one of the two terminals (Source or Drain) go negative to ground the PN junction (diode) between any of them them and the body becames positive polarized and the current will flow between terminal and body.
To avoid the problem, if the signal can go negative, the body must be held at the lowest possible negative value.
The following phrase in the book is clear in that:

If you want to switch current signals that are of both polarities (eg
-10V to 10V) ... the body terminal should be tied to -15V