Double quotes within php script echo

You need to escape ", so it won't be interpreted as end of string. Use \ to escape it:

echo "<script>$('#edit_errors').html('<h3><em><font color=\"red\">Please Correct Errors Before Proceeding</font></em></h3>')</script>";

Read more: strings and escape sequences

use a HEREDOC, which eliminates any need to swap quote types and/or escape them:

echo <<<EOL
<script>$('#edit_errors').html('<h3><em><font color="red">Please Correct Errors Before Proceeding</font></em></h3>')</script>
EOL;

Just escape your quotes:

echo "<script>$('#edit_errors').html('<h3><em><font color=\"red\">Please Correct Errors Before Proceeding</font></em></h3>')</script>";