I don't know where the "moving light" came from, certainly not from the initial quote. It is true that time durations can be measured, unless you are saying days, weeks and years are also an illusion. And it is true that an infinite duration would be required for an infinite past.

In the meantime, will one of you "other observers" fetch me a dinosaur egg and tell me a good stock to own 10 years from now?

If time exists is there a particle that transmits it?

There is no "time particle" as there is no "position particle" or "velocity particle". On a level of classical theory each force that exist in nature is mediated by some field (like electromagnetic field), in the quantum theory the fields can be described in terms of particles. So there are particles which correspond to four fundamental forces (also called interactions) observed in nature: photon for electromagnetism, bosons Z and W for weak interaction, eight sorts of gluons for strong interaction and graviton for gravity (the last one being still hypothetical as there is no consistent quantum theory of gravitation so far).

excuse the dumb question, if there is no observer does time still exist? Is time something created by our congitive processing to put order into the signals our senses take in. In other words do we create time in our head? If time exists is there a particle that transmits it?

No, time is not any more constructed than the other physical quantities we observe. We usually construct the language we speak about the nature (this includes the definition of how we understand the word "time"), but if there were some aliens describing the nature in some completely distinct way, our and their description should be completely equivalent. The nature also should behave in the same way without any observers within it (at least if you believe in existence of external reality, what seems to be not believed by some postmodernist philosophers who however usually have very sparse knowledge of physics).

So two guys 10 billion light years away decide to start moving; one moves towards us at 10MPH, the other moves away at 10MPH. These guys are merely moments away from each other yet 300 years apart according to my observations? I don't get it.

Maybe it would be good to formulate the things more precisely. First, it is better to speak about the time distance of events instead of objects (I mean it is far from obvious what does "two guys 300 years apart" mean). Imagine for example that these moving guys send a signal (this defining two events: E1 = guy no.1 pushing a button on his transmitter, E2 = guy no.2 doing the same thing on his apparatus) and imagine that from your point of view E1 and E2 are simultaneous (since the guys are placed symmetrically with respect to you you will receive both signals at the same time). The point is that from the first guy's point of view E1 happens 300 years after E2 and from the second guy's point of view E1 happens 300 years before E2.

Note also that it is not so easy for the first guy to measure the actual time of E2. Either he knows what the distance to the second guy is (and as the signals propagate with the speed of light he can compute the time needed by the signal to travel from the point where it was released) or he has to have a companion who:

1) has his clock synchronised with guy no.1
2) has zero velocity with respect to guy no.1
3) at the moment when guy no.2 pushes the button (event E2) is (almost exactly) on the same place as guy no.2 is

In such situation if the companion observes E2 at time t, first guy would observe E1 at time t+300.

These guys are merely moments away from each other yet 300 years apart according to my observations? I don't get it.

No, the two people are not 300 years apart according to your observations. But their observations of your position in time differ by a total of 300 years.

I believe the example assumes that according to your observations, the other guys are at that position simultaneously with you. Their observations disagree with you on that -- they think you're forward or behind.

Getting used to the idea that time differences in one frame of reference can never be simply transferred into another frame of reference takes a while, because we're so used to our ordinary idea of time that we have to learn to notice we're using it.

If you want to get the hang of things, I suggest Minkowski diagrams, a lot of practice, and an automatic understanding that the first three 'paradoxes' you come up with will merely be flaws in your understanding. Oh, and Brian Greene, of course.

These guys are merely moments away from each other yet 300 years apart according to my observations? I don't get it.

The analogy Brian Greene uses is that your perception of "now" can be likened to a "slice" through space-time. When we're right next to each other and moving at the same speed, our slices overlap; we agree on what is happening now. But if I'm in motion relative to you, my slice is at an angle relative to yours. It will encompass events that are either in your past or in your future, depending on which way I'm traveling.

Normally, the way to see a substantial difference is for me to travel very quickly relative to you, at some appreciable fraction of light speed, so that my now-slice of spacetime will be steeply angled compared to yours. But even a very slight angle (a very low relative speed) will do if I'm far away from you, so that the difference can accumulate over distance.

This is, of course, a very poor analogy to describe something that can only truly be described mathematically, which is what the special theory of relativity does.

I'll put up the same disclaimer: I'm not a physicist, heh, obviously.