Is $x^2\geq 0$ an axiom? For real numbers, $x^2\geq 0$ is always true, but why actually?
Is it an axiom, definition or is there a proof?

Any of the above, depending on exposition.

There are two typical axiomatic approaches to ordered fields, which boil down to how to relate multiplication to the ordering.
One version axiomatizes the ordering, and including the requirement that if $0 \leq a$ and $0 \leq b$, then $0 \leq a \cdot b$.
One can then use this, along with trichotemy, to show that $0 \leq x^2$. (using the fact $x^2 = (-x)^2$)
In the other version, one axiomatizes the positive numbers, including the requirement that $x^2$ is positive for every nonzero $x$.
