No they don't! I have eggs in different pitches, which I select to suit the song. I also modulate the EQ by cupping and opening my hand round them.
As I said before, the advantage of the egg is that being small, there is only a very short delay in the gravel inside crossing the space inside, so you can play the thing in time. I have a big Jamaican gourd shaker which sounds great, but plays up to a quarter of a second late - and the delay varies, making it impossible to play in time.

Cheers
Dave