Have you tried Text::Balanced? Or, if quotes can't be escaped, something like (untested):

while (my ($var) = $line =~ s/^(\s*['"]+=//) {  # eat single var
  # eat a single quoted value and its trailing whitespace.
  $line =~ s/^'([^']+)'\s*// || $line =~ s/^"([^"]+)"\s*// ||
    die "invalid";  # or return, whatever you need
}
# if we haven't consumed all our input, this line is invalid.
die "invalid" if length $line;
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