Which one of the following is grammatically correct?
Abc Country might be on the verge of a shutdown.
Abc Country might be on the verge of shutdown?.
I think the second one is right.
Both are grammatical, but there is a slight rhetorical difference.
Bare shutdown, with no determiner, is a state: it signifies an enduring property of existence. Using the word this way invites your hearers or readers to consider what it will be like to be in that state. It is intended to evoke alarm and distress: “Oh my goodness, will planes stop flying? Will the parks be closed? Will I not get my tax refund?”
A shutdown is an event: it signifies entering a new state of being, not the state itself. Using the word this way invites your hearers or readers to compare other events of the same sort, to consider how the event may be forestalled or if it is not, what counter-event may end it. “Yeah, yeah, 96, Frooty Nooty, Lookout Mountain was closed twenty days, been there done that.”
War works exactly the same way. Freedonia and Sylvania are on the verge of war evokes the carnage and sacrifice of warfare; Freedonia and Sylvania are on the verge of a war evokes diplomatic efforts to avert the war and the consequences of victory or defeat.
Either of the forms you propose can be grammatical, depending on how you interpret shutdown.
To be on the verge of something, as to be on the brink, on the cusp, or on the threshold, is to be very near to a point, but still outside it.
If shutdown is intended to refer to a state or an activity, like victory or crying, then no article is required. This isn't uncommon, at least here in Washington; the government is in shutdown during a shutdown, for example, this shutdown.
The species is on the verge of extinction.
The star is on the verge of exploding.
But if shutdown refers to a particular event or instance of an activity, then the article is appropriate.
Scientists are on the verge of a breakthrough in cancer research.
The stock market is on the verge of a Q3 rebound.
But there are many cases where either form would be acceptable, shutdown among them.
The currency is on the verge of collapse.
The currency is on the verge of a collapse.
I think it should be either
1) on the verge of shutting down.
2) on the verge of a shutdown.
I would prefer the first one, but I think both are possible.