Or ∃x∃y ( ~(x=y) & ∀z ( z=y or z=x) )

Still, that's not 'twoness'. That's a sentence that's only satisfied when there are two things, and could be taken as a definition of what it means to assert that there are two things, or even as a definition of there being two such things, but it's not 'twoness'. 'Twoness' implies number is a property of objects, which I think Frege pretty conclusively disproved.