The Constitution binds the US government in all cases. It does not cease to apply when the US government deals with foreigners. If the Constitution places an unqualified restriction on the US government, that restriction applies in all circumstances. The free speech clause does not have any qualification about only applying inside the borders of the US or to citizens.
It would be absurd for the US government to claim that Assange is bound by US law, and can therefore be charged under US statutes, but that the Constitution does not apply to him.