EU and NATO
A so-called "EU Army" wouldn't replace NATO, but rather complement it, and it's a post-Brexit effort to further eliminate the national sovereignty of the remaining EU members.
Such an entity would place domestic security concerns in the hands of Brussels, thus meaning that actual pro-democracy demonstrators (in this case, those agitating for an exit referendum in their home countries or a democratic reformation of the EU) could be violently put down by another member state's military forces or a coalition thereof if Brussels felt like the most government's police forces weren't doing enough to "secure the EU's interests".
In a structural (but not necessarily normative) sense, it wouldn't be much different than when the Moscow-led Warsaw Treaty crushed the anti-communist uprisings in Hungary and Czechoslovakia, although this time it would be a Brussels-led bloc doing the same thing to countries that revolt against the EU.
Objectively speaking, the move towards an EU Army can only be seen as a desperate dictatorial attempt by the Eurocrats and their American patrons to keep the remaining EU members together in more or less the same type of union as before.
This is last-ditch move by the US and it's European leadership proxies to prevent an outright decentralization of the organization by the EuroCautionary (popularly maligned as "EuroSceptic") forces in Eastern and Central Europe (led by Poland and Hungary), or, as some predict, the entity's actual dissolution.
Should the latter scenario come to pass, then the EU will likely split into several regionally focused blocs and the US would be forced to adapt its divide-and-rule strategy to these changed circumstances, which is in all actuality a "strategic inconvenience" that the US would rather not have to deal with at this time, ergo it's advocacy for an EU Army to preempt it.