The code does not act the same with or without the next. Whoever asked you the question wants you to understand how next changes the flow of the program.

for (my $i = 0; $i < @{$mydata}; $i++) {

  $account_number     = $mydata->[$i]{'account_number'} || '';
  ....

  if($email eq '') {
    bad_email("No email address found '$account_number'.");
    next;
  }
  ...
  if(($name eq '') || ($account_number eq '')) {
    $msg = "Missing information: Number = '$account_number', Name = '$
+name'.";
    next;
  }

  # Send  emails
  send_email( send my stuff here);
  ...

  exec_single_sql($sql, $db) || die "Unable to update table";
}
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This is giving you less indentation and you don't have to worry about covering error cases in end code (email, SQL).

On the other hand, I've read Dijkstra arguing that multiple exit/repeat points make it more difficult to comprehend and analyse the loop.

-- Roman

next makes it harder (or impossible) to prove your program correct, and Djikstra was all about proof of correctness.

I personally think the way next is employed can make a huge difference in readability and understandability. I would rather be able to say next if this; next if that; next if the_other; and then the rest of the loop; I find that much easier to understand.

I don't care much whether I can prove it if I can read it.

Write Perl not C (see the for loop below). Use strictures and lexical variables. Use early exits (that's yes use next to your question btw). Oh, and use place holders in your SQL:

for my $datum (@{$mydata}) {
    my $account_number = $datum->{'account_number'} || '';
    my $name           = $datum->{'name'}           || '';
    my $email = $datum->{'email'};

    if (! $email) {
        bad_email("No email address found '$account_number'.");
        next;
    }

    if ($email !~ /^[A-Z0-9._%+-]+@[A-Z0-9.-]+\.[A-Z]{2,4}$/i) {
        bad_email(
            "Invalid email address - '$email' on account '$account_num
+ber'.");
        next;
    }

    next if ($name eq '') || ($account_number eq '');

    # Send  emails
    send_email("send my stuff here");
    my $data = qq(<tr><td align="left">$account_number</td><td>$email<
+/td></tr>);

    push @all_data, $data;

    my $sql = "update table set date = ? where number = ? and date = ?
+";
    exec_single_sql($sql, $db, $datetime, $account_number, $datum->{da
+te}) 
        or die "Unable to update table";
}
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