To answer this question, it is necessary to be more precise about the meaning of "true" and "false".
In mathematics, we always work in some theory $T$ (usually ZFC), in which we can prove things.
So there is no ambiguity about formulae being provable or unprovable.
If the theory is consistent (which we hope), there is no statement $A$ such that both $A$ and $\neg A$ are provable.
However, Gödel showed that there are some statements $A$ with both $A$ and $\neg A$ unprovable (in most mathematical theories). In this case we say that $A$ is undecidable.
In this case, what does it say about $A$ being true or false?
To give a meaning to this, it is necessary to understand the notion of model. A model is a mathematical structure in which our theory is valid (i.e. all its axioms are verified).
It is only in a model that we can say that every statement is either true and false. If we stay with our theory, only "provable" and "unprovable" make sense.
In particular, if $A$ is provable, it means $A$ is true in all the models of our theory. The converse also holds, this is the completeness theorem of Gödel: if $A$ is true in all models of $T$, then it is provable in $T$.
So if $A$ is undecidable, it means it is true in some models and false in others. So the statement does not have a truth value until we choose a model to evaluate it.
What Gödel showed is that in theories that are sufficiently expressive, we can define a statement that says "I'm unprovable", because provability can be reduced to mathematical operations, and has a concrete meaning even if we only know the theory.
However, it is impossible to express "this statement is false", because "false" does not mean anything in the theory, we need to refer to a model to express it. This is why your paradoxical statement is not a well-defined mathematical statement.
ADDED (following MJD's comment)
Now, when defining a theory, we usually have a model in mind. For instance if you take Peano's arithmetic with language $(0,successor, +,\times)$, you are thinking of the model $\mathbb N$ of natural numbers (called standard model of arithmetic). We could imagine that we could define a statement "I am false in the model $\mathbb N$". However, Tarski showed that is impossible, with his undefinability of truth theorem.
It could be consistently assigned either "true" or "false".
Interestingly enough, "this statement is provable" is provable (or more precisely, in a suitable formal system to which Gödel's theorems apply,
a statement which asserts its own provability is in fact provable).
See Löb's theorem
Best Answer
"$x = 1$" is not at all the same as "$x$ can be $1$". The former states in no uncertain terms that $x$ can only be $1$ and nothing else, but that is not true if you are just given "$x^2 = x$".