I figured that out last night that I would have to change || to &&. Still, it generated a parse error. Nevertheless, after many hours hacking away at it, I figured it out. Here's the solution that worked (for anyone who has had the same issue):
$values = array('textbox1','textbox2','textbox3','textbox4','textbox5','textbox6','textbox7','textbox8');
if (strlen($values[$textbox1]) == 0 && strlen($values[$textbox2]) == 0 && strlen($values[$textbox3]) == 0 && strlen($values[$textbox4]) == 0 && strlen($values[$textbox5]) == 0 && strlen($values[$textbox6]) == 0 && strlen($values[$textbox7]) == 0 && strlen($values[$textbox8]) == 0 &&)
{
echo '<div class="error_message">
Attention! You must enter a quantity for one of the options above.
</div>';
exit();
}