This works with the data you provided:

#!/usr/bin/env perl

use 5.010;
use strict;
use warnings;

my $re = qr{
    user=
    [<"]?
    ( \S+? )
    [>"]?
    ,?
    \s
}x;

while (<DATA>) {
    /$re/;
    say $1;
}

__DATA__
.... user=jtkirk, ip=[::ffff:127.0.0.1], top=0,....
.... user=<jtkirk@enterprise.com>, ip=[::ffff:127.0.0.1], top=0,....
.... user="jtkirk@enterprise.com" passwd="XXXXXX"....
[download]

Output:

$ pm_log_uname_regex.pl
jtkirk
jtkirk@enterprise.com
jtkirk@enterprise.com
[download]