I don't like atheists anymore than the next person. I am completely agnostic to all religion.
Aside from the obvious flaw that he is using completely subjective abstract concepts for his argument, in an objective and unreal way, here is the flaw in his logic:
To measure anything you must have a starting point and an ending point. The ending point can be infinite theoretically, but the starting point must be finite.
You cannot have a timeline with infinity in both directions without a starting point somewhere in the middle. Otherwise how would you know that both directions were infinite? You need somewhere to measure from to verify they were infinite. Thus making the timeline impossible; for the starting point to exist, an infinity must have passed before it. Given that you can have no point in time, from which to measure time in either direction, either eternity cannot exist, or we cannot measure. Given this conclusion how can we apply it to something or claim that it does exist?
Aside from this logical contradiction I will point out my thoughts on the matter.
Time is a human measurement of rate of change. Without any change around us, there is no time. If you take away the night and the day, the ticking of clocks, the beating of the heart and breathing then you are left with nothing, no change. Time ceases to exist, because time is only a scientific measurement of change. No change = no time.
This whole problem of time and the universe needs to be considered from a practical perspective. Do you remember a time before you couldn't remember anything? Of course you don't. Humans have an understanding of finites because we work with finites. There is no reason to assume anything is truly infinite because it would be impossible to measure in order to prove it was infinite. Therefore infinity remains an abstract an unreal concept.
Based on the evidence around us I feel safe saying everything is finite. And if anything were to be infinite, it would be impossible for me to know that it was infinite with 100% certainty and therefore could only be an assumption at best.
Just like all religions I would need faith that infinity can exist.