Both the theists and atheists that claim that either the universe or God has an infinite regression into the past are wrong. They are both wrong because space-time itself began at the Big Bang; time DOES have a beginning. To ask what came "before time" is nonsense, because the term "before time" is a descriptor that assumes a point in time prior to the beginning of time, and that is simply an incoherent concept. Without time, there is no "before".
I do have a logical argument proving that an infinite regression in the past is impossible. Please feel free to email me if you would like clarification:
1) If an infinite amount of time comes before an event, then the event will never occur (by definition).
2) If the event will occur in the future, then it must mean that there is a finite amount of time between the present and the future event.
3) From Premise 1, it means that there is a finite amount of time before any event occurs. Otherwise the event will never occur.
4) Because the present is happening now, it must mean that there is a finite amount of time before the present.
5) Therefore, An infinite amount of time (an infinite regression into the past) is not possible, due to the fact that the present is occurring.
An objection I've i seen from this is the proposal that time is cyclic. The same argument can be applied to the proposal:
1) If an infinite amount of cycles precede one cycle, than the cycle will never happen.
2) Because the current time-cycle is happening how, this must mean that there is a finite amount of time-cycles that came before the present cycle.
3) Therefore, an infinite amount of cycles into the past is impossible.
The point of this logical argument is to demonstrate that there has to be a beginning somewhere, and an infinite regress into the past is impossible, either with the proposal that time is linear or cyclic.
If God had existed for an infinite amount of time in the past, then it means that God would never be able to do anything, because doing anything in the future requires a finite amount of time before the event can happen. It also implies that there would be an infinite amount of time preceding the present, and thus the present would not happen. But because the present is happening now, it proves that there IS a finite amount of time before the present, and ergo, there is a beginning to everything.
Your question assumes that an infinite regress into the past is necessary to explain the present. On the contrary, I think that an infinite regress into the past is impossible if the present exists.
Therefore, neither the universe, or God (if such a thing existed) can have "eternal existence".