The basic issue has been raised as to how the axiom of choice is to follow from “maximality”. This has been particularly well understood for a long time, e.g., in the following way.
THEOREM. In ZF, the following are equivalent. i. Every graph has a maximal clique. ii. The axiom of choice.
It is most convenient to define a graph as a pair (V,E), where E is an irreflexive symmetric binary relation on V. A clique is a set where any two distinct elements are related by E. Maximal means inclusion maximal.
Alternatively, one can use digraphs in the sense of paris (V,E), where E is a binary relation on V. A clique is a set where any two elements are related by E. (You can also use: any two distinct elements are related by E).