Its because $' and $& are magic, and come with a performance penalty, and in general you shouldn't use them

Also, you shouldn't use them without testing for sucess, without writing  if ( $line=~/$pattern/ ){ ... }

Change  my $output= q{" $& | $'"}; and you'll get

Use of uninitialized value $& in concatenation (.) or string at (eval 
+1) line 1, <DATA> line 1.
Use of uninitialized value $' in concatenation (.) or string at (eval 
+1) line 1, <DATA> line 1.
[download]

Add  local $&; on top of your program, that is mention $& in your code, and the penalty kicks in and the warning goes away

Also, there might be a bug in there too, for example this doesn't trigger this warning

#!perl -w
$o = q{qq{ $& | $'}};
$p = q{.};
$_=1234;
/$p/;
print eval $o;
__END__
 1 | 234
[download]

But this does

#!perl -w
$o = q{qq{ $& | $'}};
$p = q{2};
$_=1234;
/$p/;
print eval $o;
/$p/;
print eval $o;
__END__
Use of uninitialized value $& in concatenation (.) or string at (eval 
+1) line 1.
Use of uninitialized value $' in concatenation (.) or string at (eval 
+1) line 1.
  |  2 | 34
[download]

On a scale of importance from 1 to 10, I'd rate this bug a -3000 :)