That's not a very good benchmark, as you will be pushing the answer onto the same array over and over again. After running your benchmark, @main::sol contains 28200 elements.

I don't understand why you are using our all over your program. What's wrong with my?

But you missed a much faster solution: just taking the squares of the numbers from 1 to the square root of 20_000.

Here's a revised benchmark:

use strict;
use warnings;

my (@sol1, @sol2, @sol3);
my $lights = 20_000;

sub sol1 {
        #init;
        my @lit = (1) x ($lights+1);
        
        #flip;
        for (2..($lights+1)) {
                my $cnt = $_;
                while ($cnt <= $lights) {
                        $lit[$cnt] = !$lit[$cnt];
                        $cnt+=$_;
                }
        }
 
        #answer;
        @sol1 = grep {$lit[$_]} 1 .. ($lights+1);
}
 
sub sol2 {
        @sol2 = ();
        for (1..$lights) {
                my $sqr = sqrt($_);
                push @sol2, $_ if int($sqr) == $sqr;
        }
}
 
sub sol3 {
        @sol3 = map {$_**2} 1 .. sqrt($lights);
}


use Benchmark ':all';
cmpthese( -10, {
        sol1 => \&sol1,
        sol2 => \&sol2,
        sol3 => \&sol3,
});     

__END__
       Rate   sol1   sol2   sol3
sol1 3.92/s     --   -90%  -100%
sol2 41.0/s   945%     --   -99%
sol3 3762/s 95923%  9085%     --
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