I read that the natural numbers ($\mathbb{N}$) are not an elementary substructure (ES) of the integers ($\mathbb{Z}$) ,
the integers ($\mathbb{Z}$) are not an ES of the rational numbers ($\mathbb{Q}$),
but $\mathbb{Q}$ is an ES of $\mathbb{R}$?
Does this have something to do with the density, "no end-point" properties?
Presumably your language $L$ is one with a single predicate symbol $\lt$, which is interpreted in your $L$-structures as the usual "less than" relation. Then yes, it has very much to do with the densely ordered, no first or last point property. A proof can be obtained by a Back and Forth argument close to the Cantor proof that any two countable densely ordered sets with no first or last element are order isomorphic.
If we consider the various structures as $L$-structures over the usual language of rings, then none is an elementary substructure of another. For one thing, no two of the structures are even elementarily equivalent.