Lets attack the easier part first:
If a sentence can say that it is false then there is a sentence x such that
1) x = "x is false".
But according to Leibniz law we then have that
2) x is true IFF "x is false" is true.
And by the definition of truth we may simplify and we get
3) x is true IFF x is false
We have contradicted the assumption so we conclude that
There is no x such that x = "x is false". (QED)
Now to the more general question:
Is there a sentence that says Z about itself?
If so then there is a sentence x such that
1) x = "Zx"
Suppose now that
Then we have by substitution that
And we conclude that
4) IF (x = "Zx") THEN (Zx IFF Z"ZX")
So a statement x can say Z about itself only if Zx has the same truth value as Z"Zx".
These ideas are my own and I assume I may post them wherever I find them relevant!
At least until I have been proven wrong.