Yes, stick with the binary search. Just change the terminating condition from

$query == $array->[$center]
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to

$query == $array->[$center] && ($begin
   ? ($center == 0        || $query != $array->[$center-1])
   : ($center == $#$array || $query != $array->[$center+1])
)
[download]

You check the condition a second and third time later down, so you need to fix those too.

All together:

sub binarySearch {
  my ($begin, $query, $array) = @_;

  return 0        if $query < $array->[0]        and $begin == 1;
  return $#$array if $query > $array->[$#$array] and $begin == 0;

  my ($left, $right, $prevCenter) = (0, $#$array, -1);

  while(1) {
    my $center = int (($left + $right)/2);

    my $cmp =  $query <=> $array->[$center]
            || ($begin
                 ? ($center == 0        || $query != $array->[$center-
+1] ? 0 : -1)
                 : ($center == $#$array || $query != $array->[$center+
+1] ? 0 : +1)
               );

    return $center if $cmp == 0;
    if ($center == $prevCenter) {
      return $right   if $begin == 1;
      return $right-1 if $begin == 0;
    }
    $right = $center if $cmp < 0;
    $left  = $center if $cmp > 0;
    $prevCenter = $center;
  }
}
[download]

Update: Added code.