It is not possible to know the variable name of the variable passed into the subroutine, you need to tell it the name of the varaibles.

Just a minor modification to your code will make it work - just pass in the variable name in csv format.

#!/usr/bin/perl
use strict;
use warnings;

sub defnz  {
    my @names = split /,/, shift;
    my %vars;
    @vars{@names} = @_;

    for my $item (@names) {
        if (!defined($vars{$item})) {
            print STDERR "$item is not defined\n";
            return;
        } elsif ($vars{$item} eq '') {
            print STDERR "$item is empty\n";
            return;
        }
    }
    print STDOUT "All scalars defined and non-empty\n";
    return 1; # all items in list defined and non-empty
}


my ($foo,$bar,$baz) = ("FOO","BAR","BAZ");
defnz('$foo,$bar,$baz', $foo,$bar,$baz);
($foo,$bar,$baz) = ("FOO","BAR","");
defnz('$foo,$bar,$baz', $foo,$bar,$baz);
($foo,$bar,$baz) = ("FOO",undef,"");
defnz('$foo,$bar,$baz', $foo,$bar,$baz);

__END__
output is:

All scalars defined and non-empty
$baz is empty
$bar is not defined
[download]

Simpler still, you can use Data::Dumper to inspect the variables.

#!/usr/bin/perl
use strict;
use warnings;
use Data::Dumper;

sub defnz  {
    my @names = split /,/, shift;
    my %vars;
    @vars{@names} = @_;

    for my $item (@names) {
        if (!defined($vars{$item}) || $vars{$item} eq '') {
            print STDERR Dumper(\%vars);
            return;
        }
    }
    print STDOUT "All scalars defined and non-empty\n";
    return 1; # all items in list defined and non-empty
}


my ($foo,$bar,$baz) = ("FOO","BAR","BAZ");
defnz('$foo,$bar,$baz', $foo,$bar,$baz);
($foo,$bar,$baz) = ("FOO","BAR","");
defnz('$foo,$bar,$baz', $foo,$bar,$baz);
($foo,$bar,$baz) = ("FOO",undef,"");
defnz('$foo,$bar,$baz', $foo,$bar,$baz);

__END__

output:

All scalars defined and non-empty
$VAR1 = {
          '$foo' => 'FOO',
          '$baz' => '',
          '$bar' => 'BAR'
        };
$VAR1 = {
          '$foo' => 'FOO',
          '$baz' => '',
          '$bar' => undef
        };
[download]