As far as I can tell (I have never watched the movie), the shuttles depicted in Armageddon are significantly different from the real-world Space Shuttle -- they're clearly supposed to be a different vehicle. Additionally, Armageddon is an extremely unrealistic Hollywood movie that doesn't make the even the slightest attempts at accuracy common in hard science fiction.
There are some pretty significant problems with adding external rockets to the shuttle to provide large amounts of additional delta-V (since the OMS system, even with the proposed-but-never-built add-on tank kit, was very weak).
First, the Shuttle is very sensitive to balance and aerodynamics -- and it must fly stably through the whole aerodynamic flight regime. On a vehicle as fragile and high-performance as the Shuttle, it's pretty sensitive to disruption. It's likely that any attempt to add external rockets would be a massive engineering project unsuitable for an emergency effort. The mounting points would also be problematic since they must pierce the thermal protection system, and because they look like they would prevent the cargo bay doors from opening, which is necessary since the radiators are on the inside of the doors.
Second, mass is an issue. The Space Shuttle orbiter weighs about 78 tons, and has a cargo capacity of about 23 tons to Low Earth Orbit (and a significantly lower capacity to higher or inclined orbits). To get to the Moon (orbit) from Low Earth Orbit one needs about 3600 m/s of delta-V. The best specific impulse chemical fuel available (discounting theoretical ultra-dangerous ones involving fluorine that nobody ever uses) is LOX/LH2 for about 450 m/s specific impulse (notably, also used by the shuttle main engines). With those requirements, you end up needing about 100 tons of propellant, even if the boosters weigh nothing. Clearly this is not tenable -- and this doesn't even include propellant to get home.
Realistically a better idea would be to use the shuttles (assuming they're available) to launch a minimalist spacecraft which can be much, much lighter (since it does not need to survive flying in the atmosphere), use that to fly to wherever you want to go, and then use either the shuttles or a small capsule like Apollo, Gemini, Orion, Soyuz, Crew Dragon, or whatever, to return to Earth. A Crew Dragon weighs about 12 tons, and Gemini is under 4 tons (but very, very cramped). You would launch a capsule, either on a larger-than-normal rocket to provide a booster under the capsule for interplanetary flight, or launch one on a separate rocket and dock in orbit. A Falcon Heavy can launch 64 tons to LEO, which is plenty for both a Crew Dragon and a nice big upper stage to carry it wherever it needs to go, or it an instead just give the Crew Dragon a boost towards its destination (You could send it to Mars, though you couldn't send the supplies needed to survive the trip).
Dream Chaser is hard to get much detail on because it's still in development and keeps changing, plus it's private. I can't find reliable mass numbers, but it looks like it weighs about 10 tons. It is not very much like the Shuttle at all -- it's meant to be launched on the top of a big rocket, which gets it into Low Earth Orbit. Basically, the Dream Chaser could be used similarly to the small capsules I described: Launch it on a bigger than normal rocket so you have an upper stage once in space, or launch it and then launch a big rocket stage separately and dock them (the docking port is in the tail). Putting boosters on the sides of it would be awkward due to the shape, you probably couldn't put very big ones on (once again, they need to be bigger than the ship itself to get decent performance), but as it's launched as payload on top of a rocket, it likely is not quite as intractable.