How will the Boom SST not require an afterburner?
1. Will they use variable geometry exhaust?
According to Wikipedia, with references, they plan to use variable inlet/exhaust. So yes, which is similar to the Concorde.
2. Are there other ways?
In theory, the variable cycle engine (VCE), which according to them they won't go for due to the R&amp;D cost. Those engines are also heavy and very noisy, even by the 80's standards.
3. Which turbofan are they going to use?
For the scaled demonstrator they built, they are going for a turbojet, the General Electric CJ610 from the 60's. At the present time, we don't know is the right answer. But the technology can be analyzed.
The inlet of a  supersonic turbojet slows down the oncoming air so it can be compressed and then accelerated by combustion out the back. And as you hinted, a converging nozzle is needed for the supersonic regime. This is the classic accelerate a small amount of air by a lot.
The propeller and the fan (in the turbofan) are the opposite, they accelerate a lot of air by a little. And it's easy to visualize why. The optimum blade tip speed is around Mach 1.0 give or take, so the average velocity of the air behind the fan after gaining energy will not be much faster, but it will be a lot of air (also note, away from the tip it's slower). This is where I don't get it, why have an inlet that slows down the air, only to accelerate it by a little bit, instead of accelerating it by a lot?
They also note that the engine they will settle on will have higher maintenance cost, which hints at running at extreme power levels. This is to be expected as the previous paragraph shows. This answers how the no-reheat. The aim seems to be noise reduction, not economics (wait for it).
Onto some true context:
They compare a fare of \$5,000 to the Concorde's \$20,000 (adjusted for inflation). What they don't say is that the Concorde's original fare was around £1,000, and it only increased because the management realized the rich who flew the Concorde didn't even know how much they were paying.
Adjusted for inflation from 1981, we are looking at an original fare of \$5,500. The reason is simple, to this date, the Concorde's engine remains the most efficient, when you factor in the speed.
Note: I was being very conservative when I said £1,000, in 1980 it was actually £600. And BA was reporting profits on the NYC route.

In short, they already said they can't afford the R&amp;D of the VCE technology, so don't expect something new, it will be present day transonic technology used in the wrong flight regime, which explains the high fare.
Note on the Concorde afterburner:

The reheats are turned off shortly after take off. The reheats are turned back on, for around 10 minutes to push the aircraft through Mach 1 and on to Mach 1.7 where they are no longer required.

Addressing comments:

I would think that higher maintenance costs could be completely attributable to the small market that this engine will have.

If they don't have financing for VCE R&amp;D, then they should settle on an engine with parts commonality to existing models to reduce the maintenance cost.

Existing turbines in reliable daily operation all around the world make enough [thrust] to power the Boom jet.

In the supersonic regime, if they don't use a turbojet or at the very least, a low BPR turbofan, they're wasting fuel, but yes, they will be keeping the noise down. I just didn't like the false marketing of a low-fare. Engine manufacturers would not have been chasing VCE's if medium BPR was the solution all along. The problem with supersonic flight and the low speed of the turbofan exhaust can be understood from this NASA page. The faster the freestream, the faster the exhaust needs to be.