In a field, an element can be made negative in some order iff it is not a sum of squares. Thus, $F$ is uniquely ordered iff for every $a\in F^\times$, $a$ or $-a$, but not both, is a sum of squares.
An order on a domain $R$ extends uniquely to its fraction field, thus $R$ is uniquely ordered iff its fraction field is. Thus:
Proposition: The following are equivalent for any domain $R$:
- $R$ is uniquely ordered.
- $0$ is not a sum of nonzero squares, and for every $a\in R$, there is a nonzero $b\in R$ such that $ab^2$ or $-ab^2$ is a sum of squares.