Do light particles have thrust? I understand that nothing is faster than light and that it can not escape a black hole. However, light particles may be fast, but perhaps it can't escape a black hole due to it's lack of thrust power? I can't reasonably push an object with light. a rocket has thrust but can't go as fast as light and light has speed but can't go through sheetrock. It just doesn't seem that light has much strength to it. Quasars, spew light out due to a force pushing light out.  Can this be explained to me?
 A: Light has momentum given by $$p=\frac{E}{c}$$ where $E$ is the energy of the photons and $c$ is the speed of light.
And light can indeed push objects since as the photons strike the surface, the momentum of the photons are transferred to the surface of the object. You can actually get "thrust" by exploiting this fact. And a solar sail is one  method of doing just that. From that link:
"Solar sails (also called light sails or photon sails) are a method of spacecraft propulsion using radiation pressure exerted by sunlight on large mirrors. A number of spaceflight missions to test solar propulsion and navigation have been proposed since the 1980s. The first spacecraft to make use of the technology was IKAROS, launched in 2010."
A Crooke's radiometer also demonstrates the fact that light can move things. It has vanes mounted on a spindle and when light is shined on these vanes they start to rotate, and the higher the intensity of the light the greater the rate of rotation.
As for the other part of your question, light cannot escape a black hole beyond the event horizon due to the extreme curvature of the spacetime. In technical terms, all null geodesics in a black hole end at the singularity which simply means that all paths that light can take inside event horizon remain inside the event horizon.
A: One of the weird things about light is that in vacuum it always moves at the speed of light through space-time, which means that there is nothing we could do to make it go faster. So if you were to give thrust, you could sort of see that as giving a photon more energy I suppose, but, if we consider the simplest case of a neutral, non-rotating black hole, when we calculate the trajectory the photon takes, and thus ultimately whether or not the photon enters the black hole or not, the energy of the photon doesn't come into play at all, just the simple fact that it travels at the speed of light. I suspect the same holds for charged and rotating black holes, but I don't have the specialism to comment with confidence.
Now that I've addressed what I think you want to know about, I'd like to talk about a few misconceptions in your question: when we say nothing can escape a black hole, we really mean nothing, no matter how hard it is thrusting, once it has crossed the event horizon. The reason for this is that space-time is curved very strongly there - imagine it like a car on a hill, at some point, no matter how good the car is, it can't get up the hill - this is kind of the same thing, but it's gravity that represents the hill, and no matter how hard we push, once we're inside the event horizon, we can do nought but go towards the centre of the black hole.
Quasars don't actually push light out either - they emit extremely bright light due to the insanely energetic physical processes that occur near quasars, but they actually "pull" on the light a bit, due to them curving space time. This redshifts the emitted photons frequency as they get further away from the quasar : you can think of it as curved spacetime bunches up the wavelength more than flat spacetime.
A: As pointed out by joseph, light has indeed a momentum.
Its extremely small, still it is measurable. The origin of this property can be found in the wave particle dualism of electromagnetic radiation.
In space, the momentum of photons is even being utilized as a form of repulsion based thrust with the help of solar sails. This is possible, because in space there is no air based drag. There are also concepts, in which space vehicles are getting their thrust by focused laser beams. When the mass of these objects is reasonably small, they could in theory reach a significant portion of the speed of light itself.
For further reading on this:
https://en.wikipedia.org/wiki/Solar_sail
