I am confused also. This appears to be a very contrived example. Can you show some code closer to your actual application?

For your regex, just keeping track of last vs curr pos would seem to do it. $ matches when pos does not advance.

use warnings;
use strict;

my $str = "abc";
my $last_pos =0;
while ($str =~ m/.|$/gc) {
   my $curr_pos = pos($str);
   printf("1: %d", $curr_pos);
   ($curr_pos == $last_pos) ? print "  EOString\n" : print "\n";
   $last_pos = $curr_pos;
}
__END__
1: 1
1: 2
1: 3
1: 3  EOString
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For character by character processing of a string, substr() and it's buddies are appropriate, not regex. An example showing more of what you are really trying to accomplish would be helpful.