TL;DR: a solution!
[FEN "rnbq1bn1/p2pp1p1/1p2r1Nk/2p4p/6PN/1BPP1p1P/PP2PP2/R1BQK2R w - - 0 1"]
32 pieces, no squares.
According to OEIS sequence A240443 and this particular pair of examples, 34 is the maximum possible number of square-free points on an 8x8 grid. As long as each file has at least two points in ranks 2-6, this is likely to give us a solution, if we just drop a couple of the points. (I picked two from the more crowded files.)
This is not intended to be optimal in number of moves - it just shows that a capture-free solution is possible, and is a starting-point for the search for the shortest (non-unique) proof game (which we now know lies somewhere between 4.5 & 7.5 moves).
The rest of this post gives more details.
Generalization to abstract math problem
Overall, I make it 336 possible “squares” on the chessboard, of which 140 are non-tilted and 196 tilted. Since about 60% are tilted, avoiding the tilted squares in a general position is the harder challenge, although there were only 2 tilted squares to avoid in the game array.
Following an exploration into positions where ranks 3-6 were empty, I think the best general approach may be to forget about the chess for a while. We've got 64 nodes, and 336 "quartets", each of 4 nodes. What's the smallest subset of 64 which contains a member of each quartet? There is no a priori reason why that smallest subset should be greater than 32. (I.e. we might be able to have more than 32 pieces on a square-free board!) Anyway, after that smallest subset is determined, we can see how we can arrange chess pieces to avoid those squares on the board.
Having got a solution, the next thing might be to determine all maximal arrangements of 32-34 square-free points which allow for 2 pawns/file. (It’s the 34-pointers which are most interesting, although it’s possible that some position with 32 or 33 point might not be extensible to 34, so would be missed in any trawl of 34-pointers.) Can then look at the smallest number of moves to reach positions which have 32 pieces. In such open capture-free positions, it seems very unlikely that a unique proof game exists.
If anyone does get a list of these maximal 32-34 point diagrams, I would be grateful if they can be posted in GitHub. There is some python code in OEIS, I saw.
Bounds for shortest (non-unique) proof games
An earlier reconnaissance to look at homebase positions (i.e. where all surviving units are on their apparent starting squares) proved that a maximum of 23 units can remain on the board. A valid implication for the general problem is that any (non-unique or unique) proof game to a square-free position must have at least 4.5 moves (i.e. White moves at least 5 times). This bound is unlikely to be attainable.
Elsewhere in this thread, there is a solution in 7.5 moves. Any faster solution must therefore have at least 9 units on ranks 1 & 2, and at least 9 units on ranks 7 & 8. This constrains somewhat the search for interesting 32-34-pointers.