How could a small, civilian helicopter handle a mounted M134 Minigun? Pretty much the same way this guy fires one without a mount, with the gun attached to nothing but himself (skip to 2:20):
The weight of the gun and chopper tend to counteract the recoil of the much smaller projectile and explosions.
As this answer pointed out, the fact is that a handheld minigun is impractical, dangerous, and difficult, but certainly possible.
I haven't seen the movie, so I don't know if this is the right one, but several sources suggest that the chopper is a Eurocopter EC130. It was heavily modified for the movie, and the specifications of even stock helicopters in the line vary from model to model, but here's an example of what they can do:
Empty Weight: 3,036 lbs
Useful load: 2,315 lbs
Max. takeoff weight: 5,351 lbs
The IMFDB says the chopper was actually an EC120 Colibri, also manufactures by Eurocopter. Here are the specifications for that helicopter:
Empty weight: 2,185 lbs
Useful load: 1,596 lbs
Max. takeoff weight: 3,781 lbs
Less beefy than the 130, but not by much.
So far as I can find, although both of these aircraft are used by multiple militaries around the world, neither is typically armed. In fact, I found no evidence of either one being equipped with weapons.
Helicopters in General:
The rear rotors of helicopters (which spin vertically, like a table saw) are designed to keep the force generated by the main rotors (which spin horizontally, like a ceiling) from spinning the entire aircraft in circles. The energy put out by these rotors is far greater than the recoil of a minigun.
On top of this, the main rotor acts as a gyroscope. If you've ever played with a bicycle wheel, holding one side of the axel in one hand while spinning the wheel with the other hand, you know that the wheel becomes far easier to keep upright while it is spinning. That's a gyroscope at work. Helicopters do the same thing. There's a very good reason why we use gyroscopes to stabilize stuff.
If a helicopter can handle the stress exerted by the rotors, it can certainly handle a bit of recoil from a minigun.