You have to set the text rendering mode to 2  (2=fill+stroke — default is 0=fill only):

...
my $headline_text = $page->text;
$headline_text->font( $font{'Helvetica'}{'Bold'}, 18 / pt );
$headline_text->fillcolor('green');
$headline_text->strokecolor('red');
$headline_text->linewidth(0.2 / mm);
$headline_text->render(2);                      # <--
$headline_text->translate( 95 / mm, 15 / mm );
$headline_text->text_right('My Headline');
...
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Yes, the docs are a little terse...: "$rendering = $txt->render $rendering" — so you're not to blame for not finding it :)

(See also section 5.2.5 "Text Rendering Mode" and Table 5.3 (p. 402) in the PDF Reference v1.7.)