Generally, a topos over a base topos is called an atomic topos if is atomic.
See (Johnstone, p. 689).
Atomic morphisms are closed under composition.
An atomic geometric morphism is also a locally connected geometric morphism.
This appears as (Johnstone, lemma 3.5.4).
This appears as (Johnstone, cor. 3.5.2).
If logical then it preserves the isomorphism characterizing a Boolean topos and hence is Boolean if it is atomic.
For the converse…
This appears as (Johnstone, lemma 3.5.4 (iii)).
Every etale geometric morphism is atomic.
Another example of an atomic Grothendieck topos is the Schanuel topos.