The object is to get the shortest possible string that starts with " and ends with either " or the end of the sample. So:

while (<DATA>) {
    chomp;
    print "$1\n" while m/(".*?(?:"|$))/g;
}

__DATA__
abc
"def
"ghi"
jkl"
mno"pqr"stu
v"wx"y"z"
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