A one-liner regex is not the most maintainable or efficient solution to this problem. But since you asked, here's a regex answer:

s/((?:^|\|)(?:[^"|]*"[^"|]*")*[^"|]*)"([^"|]*)(?=$|\|)/$1$2/g; [download]

The readability of that could be improved upon, however:

my $T = qr/[^"|]*/; s/((?:^|\|)(?:$T"$T")*$T)"($T)(?=$|\|)/$1$2/go; [download]

Now here's how I would have really done it:

join '|', map { s/"([^"]*)$/$1/ if y/"// & 1; $_ } split /\|/; [download]

   MeowChow                                   
               s aamecha.s a..a\u$&owag.print