So a minor change to allow the closure to close on the current value would be to create new variables:

#! /usr/bin/perl 
use warnings;
use strict;
use Tk;

my $mw = MainWindow->new;
$mw->title("Checkbutton Grid");

my $x_size = 5;
my $y_size = 5;
my %b_name;

for (my $x=0; $x<$x_size; ++$x) {
    for (my $y=0; $y<$y_size; ++$y) {
        my $x_cord = $x;
        my $y_cord = $y;
        $mw -> Checkbutton (
                            -activebackground => "white",
                            -command => sub { print "$x_cord,$y_cord\n
+";
                                print "hi"; }
                           ) -> 
            grid (
                  -row => $x,
                  -column => $y
                 );
    }
}

MainLoop;
[download]

I'm not saying this is more efficient or better than your other example below (probably is neither), just that it's closer to the OP tried to do while still working.