The below might be good enough for your purposes:
it splits up $screen into an array of single chars (using the split // idiom to split a string into each character), then loops through the resultant array to find non-X characters that are either

• 1) at the start of the array, and therefore a starting point, or
• 2) just after an X character, and therefore a starting point.

#! /usr/bin/perl -w
use strict;

my $screen = "ATCGATCGXXXXXATCGATXXXACTGCTACGGTACXXXAATTATXGCGCGXXT";
my @chars = split //, $screen;
my @starts;

foreach ( 0 .. scalar @chars - 1 ) {
    push @starts, $_
        if $chars[$_] ne 'X' &&  ( 0 == $_ || $chars[$_-1] eq 'X' );
}

print "$screen\n@starts\n";
[download]

Try checking out the bioperl modules, at http://search.cpan.org/dist/bioperl/, also. I'm not familiar with them, but they have a prepackaged routine for this.