<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.2.1/jquery.min.js"></script>
<textarea name="text" class="form-control" rows="14" id="text" placeholder="Here goes our query"></textarea>
<input id="clear" type="button" value="Удаление пустых строк">
<script>
$(document).ready(function () {
$('#clear').click(function () {
var text = $('#text').val();
text = text.replace(/(?:(?:\r\n|\r|\n)\s*){2}/gm, "\r\n");
$('#text').val(text);
});
});
</script>