How long does a non-ETOPS flight need to be able to fly in an one-engine-out situation?
No, nothing addresses that case.

Let's say you're 20 minutes out from your destination, and because it's a clear VFR day you have minimum normal reserves -- roughly an hour of flying (on 2 engines, at medium-to-low altitude).  Those reserves are already prescribed for you by FAR's.  Now you lose the engine.  How long can you fly?  Until you run out of fuel or the other engine quits -- so in all probability, about that same hour, plus or minus.

Different case, you're over at Atlantic heading to Bermuda or the Azores or Iceland or NYC to the Caribbean, or over the Gulf of Mexico -- not ETOPS but a long ways from anywhere.  The fuel on board will be lots more, because you were planning for a much longer flight.  (And again, those fuel requirements are defined elsewhere.)  You now have the added twist of drifting down on one engine to a single-engine ceiling -- you set max continuous thrust on the good motor, and descend at a speed that gives you max range, which probably comes out to a few hundred feet per minute of descent.  Back in the flight planning, the check had to be performed that at the worst case, you could lose the motor, drift down, and continue to reach a suitable airfield without needing more gas than you'd have.  But in something like a 737, the gas you'll have on-board anyway is almost certainly more than sufficient.

But in this latter case, the time you can stay aloft is going to be hours -- way beyond what you had in the first case.

So no, the regulations don't prescribe a time that you have to be able to keep flying after losing one engine.  You DO have to be able to keep flying -- that's important -- and you have to have enough fuel to get to a destination with the one motor shut down.  But in terms of a prescribed time, that isn't given in the FAR's.