Here's one that lets the regex engine find the largest square all by itself :)

#!/usr/bin/perl

# https://perlmonks.org/?node_id=1233341

use strict;
use warnings;

my $original = local $_ = do { local $/; <DATA> };
s/.\K  //g;                       # tweaks to make it easier to work w
+ith
my $width = /\n/ && $-[0];
s/.+/sprintf "%-${width}s", $&/ge;
my @squares;

while( s/x| / / )
  {
  my $pos = $-[0];
  pos($_) = $pos;
  /\G( +)(??{ ('.{' . ($width + 1 - length $1) . "}$1") x ~-length $1}
+)/s
    or die "tricky regex failed - oops";
  my $size = $& =~ tr/\n// + 1;
  for my $n ( 1 .. $size )
    {
    substr $_, $pos, $size, ('A' .. 'Z')[@squares % 26] x $size;
    $pos += $width + 1;
    }
  push @squares, [ $pos % ($width + 1), int $pos / ($width + 1), $size
+ ];
  }

use Data::Dump 'dd'; dd \@squares;
print $original;
print s/./$&  /gr =~ s/  $//gmr;

__DATA__
0  1  2  3  4  5  6  7  8  9  0  1  2
1
2                             x
3                    x
4        x
5  x
6              x
7                       x  x
8        x
9        x                 x
0        x
1     x                 x
2
[download]

EDIT: removed leftover unneeded instructions.