Yes, you can most definitely fit an infinite number of non-overlapping successively smaller squares following your indicated pattern inside a unit square. That's what the summation you did proved. Compare the areas of the sum of the infinite squares to the area of the unit square:
i=2∑∞i−2=6π2−1≈0.6449≤1
Although this proves you can fit all the squares inside, it, as you correctly identified, doesn't tell you
how to arrange the squares. But the reality is that it doesn't matter in this case, because there are infinitely many ways to arrange the smaller squares so that they all fit. One such way is to place the first square, with area 1/4, in the lower left corner of the unit square. Then place the second square, with area 1/9, so that the lower left corner of it is at the same point as the upper right corner of the first square. Repeat indefinitely.
See if you can find some others, and hopefully in the process intuit out why there must be infinitely many arrangements. In fact, I believe (although I'm not 100% sure) that there is an infinite number of ways to arrange
any number (finite or infinite) of non-overlapping squares of
any size inside a unit square, so long as the squares' areas sum to 1 or less, and the biggest square has an area of 1/4 or less.
Maybe play around with that too and see what comes of it. I think it will further your understanding of the ideas at play here.