Cellular Automata: Could we use Conway's Game of Life in file compression?

No, because many predecessor states can lead to the same successor state. Think, for example, of all the states that lead to the death of all cells. So if you start with a successor state there is no way not just to find a unique smaller predecessor but one may not even exist.

The primary strategy of file compression is to reduce redundancy in the data.  Conway's Game of Life has no relation to redundancy elimination.

This is a very interesting question that I've thought of often. The simple answer is that since each evolved state destroys data, it could never be reversed. There is one strange thing however - When playing with the Golly simulator, I noticed one particular rule set - the amazing thing about this rule set was, given an image, after a few hundred generations, it would seem completely chaotic, but after a few more, you would get a state which had 4 copies of the original image scaled down. This would keep repeating. I recall this from 15 years ago, but this is food for thought.