Before I begin, I'd like to note that defining the cause/effect between entropy and time are a bit beyond me, so my understanding may be partially flawed.
Spacetime is a function of entropy, where an increase in entropy correlates to an increase of time. So, from this perspective, with the understanding of the laws of Thermodynamics, you can relate it to special relativity (spacetime). The "universal" maximum of entropy, also know as "heat death", would be the end of time according to this theory, is my understanding of the end of the universe. So, in order for maximum entropy to occur, all mass would be converted to light/radiation. And as you should know, time does not exist relative to lightspeed. So, if heat death is the end of the universe, it follows that minimum entropy would be the beginning of the universe. Admittedly, I don't think anyone exactly understands how or what the "beginning" even means. Since the progression of time will asymptotically approach 0, I guess it would follow that an infinite progression of time occurred at the beginning of the universe, which is a concept I'm having trouble wrapping my brain around.
See the works of Boltzmann for more details.
So, I guess my definition sort of relates to Transcix definition, however ambiguous and arbitrary his language seems.
Time and eternity are just ways of expressing human concepts. So to me it seems fair to hypothesize that before man invented the concept of time and eternity there was no time and eternity, only the balance of nature. What humans call time is just planets revolving around the sun. Since our definition of time is imperfect (think of leap years) I'd say that humans haven't even fully understood time in a universal sense yet. It can be very informative to look at the history of calenders if you're interested in the human concept of time.
If we ever encounter intelligent extraterrestrial life forms that come from a different universe it would be likely that their sense and concept of time is very different from ours. What if they're from a planet that has no solar cycle and they live to be 2000 years old for instance?
So my answer to your question is: time and eternity began existing since man came up with their concepts. And once the concepts are forgotten there will be no time and eternity anymore until some other intelligent life form comes up with a new concept.
Well, after reading both Transcix's and Fenriz's responses, Transcix does not seem very informed regarding the subject matter and has been effectively trolled hard. Knowing Fenriz, he brings about very flawed concepts that do not represent the main goal of the thread.
Calendars are measurements of time the same way we measure lengths and widths with rulers. So no, the "concept" of time is not different, but yes, the way ET measures time will be different. Yes, ET can live for over 2000 "human" years, due to special relativity.