Not to be dense, but what precisely constitutes a beginning? Is it a point in time before which something does not exist, but after which it does? If so, then what happens if time itself began at a certain point, a finite number of years ago (as in my example above). Something might have existed since the start of time, say 15 billion years ago. So the thing in question has a finite age. But there was no time in the past at which it didn’t exist, because it is as old as time. So does it have a beginning or not?
(Digression: Maybe this is why I like math better than philosophy. There, everything is spelled out in a why that isn’t open to interpretation. Although, I suppose that’s mostly because in math one is permitted to express things symbolically, rather than using English words, which often have multiple meanings.)
At any rate, I think that depending on what is meant by a beginning (and thus what is meant by eternal, since you use one to define the other) it seems possible that all the argument really proves is that something must have existed since the start of the universe, since I don’t believe there is any reason (physically speaking) to assume that there was such a thing as time before the universe existed. In fact, the phrase “before the universe existed” doesn’t really have any meaning. Someone with greater knowledge of physics than me (I have a B.S.) may come here and correct me, but my understanding is that in physics time is viewed as another dimension of the universe, much like the spatial dimensions. It is meaningless to talk about distances between objects before the big bang, when the entire universe was a single point. Likewise, it is meaningless to talk about the passage of time prior to the big bang.
I also think that you are correct to question the statement that you can’t have being from non-being. In physics, there is such a thing as a vacuum fluctuation, which is literally a particle and its antiparticle appearing out of nothing and then anihilating and returning to nothing after a very short time. Supposedly, such fluctuations are occuring constantly throughout the universe. However, the restriction on them (as I understand) is that the particles can only exist for such a small amount of time that they cannot be observed. So whether this really counts as something from nothing I’m not sure. There are occasions where they can become permanent and be observed, but in this case they must get energy from somewhere (a nearby black hole, for example), so perhaps it’s not really something from nothing.
I once read a book by a physicist (written for non-scientists) in which he suggests the whole universe may be a vacuum fluctuation, but it is unclear to me how this could be consistent with the requirement that the particles not be observed. Maybe since we’re part of the fluctuation, our observation of it doesn’t count. (At any rate, currently most physicists believe the Big Crunch won’t happen – as far as I know – due to an a still unexplained Dark Energy causing the expansion of the universe to accelerate. If the universe never collapses, it probably isn’t a vacuum fluctuation, but it was an interesting thought.)