I'm not sure whether you actually want newForm to appear in the format definition. Do you mean that you want to be able to type:
f // newForm
and get out 9? I think then you'll have to define the format like this:
Format[newForm[f[x_]]] := x^2
because the form attribute is only meant to specify built-in ones like TraditionalForm, InputForm etc.
You may instead mean:
Format[f[x_]] := x^2
which will replace in all forms.
Format[f[x_, y_, z__]] := f[x, ...]
f[a, b, c]
f[a, b, c] // TraditionalForm