The integers are the equivalence classes of differences of natural numbers, while the natural numbers are the equivalence classes of finite sets having the same number of elements (i.e. which may have a bijection between themselves), including the empty set.
How do you define the _number_ of elements of a finite set without defining natural numbers first?
You just need to be able to show an one-to-one correspondence between the elements of the two sets. If an one-to-one correspondence cannot exist, then the sets have different numbers of elements.
This relationship divides then the sets in equivalence classes. If you choose a representative of each equivalence class that you use to compare to other sets to see if they have the same number of elements and you give a name to each of those representatives, you have defined the so-called natural numbers.
This is actually how the numbers originated, for humans and for many other animals.
Nobody conceived a system of axioms and then thought about what could satisfy them. That came much later and is useful only for establishing which are the essential properties of some mathematical objects. Most of the definitions of various mathematical objects as equivalence classes correspond to their real historical origin, because recognizing that some things are equivalent according to some criterion is how abstract concepts are created based on concrete things.
When you see a red apple and a red rose, you understand that they have a common property, being red, and then you name this property "red" and you can recognize the same property in other objects.
When you see 5 sheep and 5 crows, you understand that these groups have a common property, having 5 members, and the same property characterizes the set of fingers of your hand. You name this property "five" and when you see another group of things you can compare it with the set of fingers of your hand to see if it also has 5 members.
If you don't know what a rational number is, it's the equivalence class of every pair of integers that can be simplified to the same fraction. For example, (4,6) and (2,3) are both rational numbers, and in fact are the same rational number: two thirds.
If you don't know what an integer is, it's the natural numbers, but with negative numbers.
If you don't know what a natural number is, it's either zero, or a number that follows a natural number. For example one is the number that follows the natural number zero, and two is the number that follows the natural number that is the natural number that follows zero.