Looks like everything (and everyone) points into direction of GPC and its Perl bindings. In case its license is OK (if not, whoever brave enough follow links on that page and bind to e.g. Fortran lib), but here's for a start. Your requirements are a bit odd, in that you want union if there is intersection or touch. It's solved with check for only single sub-polyline in union (assuming you neither use nor create "holes" or self-intersecting shapes). The "cleanup" is to "remove colinear points", but note that start point is not duplicated as end point.

use strict;
use warnings;
use feature 'say';
use Data::Dump 'dd';

my @sets = (
    [[0,0],[0,2],[2,2],[2,0],[0,0]],
    [[0,0],[3,0],[3,1],[0,1],[0,0]],
    [[3,0],[3,2],[4,2],[4,0],[3,0]],
    [[0,2],[0,3],[1,3],[1,2],[0,2]],
);

########################################
#use Math::Geometry::Planar::GPC::PolygonXS;

# ... skipped, don't need it

########################################
use Math::Geometry::Planar;

my $poly = Math::Geometry::Planar-> new;
$poly-> points( $sets[ 0 ]);

for ( 1 .. 3 ) {
    say "********* case #$_:";
    my $clip = Math::Geometry::Planar-> new;
    $clip-> points( $sets[ $_ ]);
    my $gpc = GpcClip( 
        'UNION', 
        $poly-> convert2gpc,
        $clip-> convert2gpc,
    );
    my @pgons = Gpc2Polygons( $gpc );
    dd $pgons[ 0 ]-> cleanup unless $#pgons
}
__END__

********* case #1:
[[[2, 2], [0, 2], [0, 0], [3, 0], [3, 1], [2, 1]]]
********* case #2:
********* case #3:
[[[1, 3], [0, 3], [0, 0], [2, 0], [2, 2], [1, 2]]]
[download]