... the negative match ...

It's important to understand that the negation is made on the result of the  /eth1\.\d{4}\@/gm match against the  $ethernet string. The match itself is actually successful! It's as if the
    if ($ethernet !~ /eth1\.\d{4}\@/gm) { ... }
statement had been written as
    if ( ! ($ethernet =~ /eth1\.\d{4}\@/gm)) { ... }
(note =~ vice !~). In fact, that's pretty much the way Perl sees the code:

c:\@Work\Perl\monks>perl -wMstrict -MO=Deparse,-p -le
"my $ethernet=`ip addr`;
 if ($ethernet!~/eth1\.\d{4}\@/gm){
     print qq{No vlans exist on device.\n};
     exit;
 }
"
BEGIN { $^W = 1; }
BEGIN { $/ = "\n"; $\ = "\n"; }
use strict 'refs';
(my $ethernet = `ip addr`);
unless (($ethernet =~ /eth1\.\d{4}\@/gm)) {
    print("No vlans exist on device.\n");
    exit;
}
-e syntax OK
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I had assumed /g was pointless on the if ...

Given what you were trying to do, it was pointless, but it was not without effect!