Judging by some responses, however, it seems as if using the term "global citizen" is seen as implicit support for some kind of central world government.
The world will not be stable with that sentiment for long. At some point, the "global citizen" will seek to express their political will; a will that will inevitably conflict with the will of the Nation-State.
To do that, the "global citizens" must have sponsors, leaders, and some way to organize. Would-be sponsors, leaders,and organizers will see the potential for gain here and will step in to sponsor, lead, and organize the "global citizens." And at that point, a central world government has been seeded and will begin to grow.
Alternately, people will remain organized in Nation-states and the "Global Citizen" mentality will fade away since it won't have any relevant meaning.
And ultimately, they'll discover that the "global human community" has as many differences as similarities and not all of those differences are compatible.