Egad is a morphed form of egads.
Egads used to
mean "ye gods" and
now is more closely linked to "OMG" or "oh god"; Some
people think there is only one god, so egad has evolved to express that idea. At the heart of it is the notion that god is a trickster who screws with us.
We use it to say that something severely messed up has just happened.