When I want to pass a variable that's local to my script into a module, I pass it as an argument to a subroutine. In your case, it would be an anonymous sub in the array-of-hashrefs:

package check_module;
use warnings;
use strict;
use Exporter;
our @ISA    = qw(Exporter);
our @EXPORT = qw(@checks);

our @checks = (
    {
        name   => 'Anybody home?',
        #script => 'qq/echo $home_dir/',
        script => sub { `echo $_[0]` },
    },
);
1;
__END__
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use warnings;
use strict;
use lib '.';
use check_module;

my $home_dir = '/home/mine';
for my $check ( @checks ) {
    chomp ( my $script_out = $check->{script}->($home_dir) );
    print "Checking $check->{name}\n\t=> '$script_out'\n";
}
__END__
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__OUTPUT__
Checking Anybody home?
    => '/home/mine'
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