How can I require at least one checkbox be checked before a form can be submitted?

Here's an example using jquery and your html.

<html>
<head>
     <script src="http://ajax.googleapis.com/ajax/libs/jquery/2.1.0/jquery.min.js"></script>
</head>
<body>

<script type="text/javascript">
$(document).ready(function () {
    $('#checkBtn').click(function() {
      checked = $("input[type=checkbox]:checked").length;

      if(!checked) {
        alert("You must check at least one checkbox.");
        return false;
      }

    });
});

</script>

<p>Box Set 1</p>
<ul>
   <li><input name="BoxSelect[]" type="checkbox" value="Box 1" required><label>Box 1</label></li>
   <li><input name="BoxSelect[]" type="checkbox" value="Box 2" required><label>Box 2</label></li>
   <li><input name="BoxSelect[]" type="checkbox" value="Box 3" required><label>Box 3</label></li>
   <li><input name="BoxSelect[]" type="checkbox" value="Box 4" required><label>Box 4</label></li>
</ul>
<p>Box Set 2</p>
<ul>
   <li><input name="BoxSelect[]" type="checkbox" value="Box 5" required><label>Box 5</label></li>
   <li><input name="BoxSelect[]" type="checkbox" value="Box 6" required><label>Box 6</label></li>
   <li><input name="BoxSelect[]" type="checkbox" value="Box 7" required><label>Box 7</label></li>
   <li><input name="BoxSelect[]" type="checkbox" value="Box 8" required><label>Box 8</label></li>
</ul>
<p>Box Set 3</p>
<ul>
   <li><input name="BoxSelect[]" type="checkbox" value="Box 9" required><label>Box 9</label></li>
</ul>
<p>Box Set 4</p>
<ul>
   <li><input name="BoxSelect[]" type="checkbox" value="Box 10" required><label>Box 10</label></li>
</ul>

<input type="button" value="Test Required" id="checkBtn">

</body>
</html>

Make all the checkboxes required and add a change listener. If any one checkbox is ticked, remove required attribute from all the checkboxes. Below is a sample code.

<div class="form-group browsers">
    <label class="control-label col-md-4" for="optiontext">Select an option</label>
    <div class="col-md-6">
        <input type="checkbox" name="browser" value="Chrome" required/> Google Chrome<br>
        <input type="checkbox" name="browser" value="IE" required/> Internet Explorer<br>
        <input type="checkbox" name="browser" value="Mozilla" required/> Mozilla Firefox<br>
        <input type="checkbox" name="browser" value="Edge" required/> Microsoft Edge
    </div>
</div>

Change listener :

$(function(){
    var requiredCheckboxes = $('.browsers :checkbox[required]');
    requiredCheckboxes.change(function(){
        if(requiredCheckboxes.is(':checked')) {
            requiredCheckboxes.removeAttr('required');
        } else {
            requiredCheckboxes.attr('required', 'required');
        }
    });
});

This should have what you need, check out the jsfiddle at the bottom:

$(document).ready(function () {
$('#txt').val($("input[type=checkbox]:checked").length);
$('#txt2').val($("input[type=checkbox]").length);

$('input[type=checkbox]').change(function () {
checked = $("input[type=checkbox]:checked").length;
$('#block').show();
$('#block2').hide();
if (checked > 0) {
  $('#block').hide();
  $('#block2').show();
  $('#txt').val(checked);
 }
});
});

<ul>
    <li><input class="checkboxes" name="BoxSelect[]" type="checkbox" value="Box 1" required><label>Box 1</label></li>
    <li><input class="checkboxes" name="BoxSelect[]" type="checkbox" value="Box 2" required><label>Box 2</label></li>
    <li><input class="checkboxes" name="BoxSelect[]" type="checkbox" value="Box 3" required><label>Box 3</label></li>
    <li><input class="checkboxes" name="BoxSelect[]" type="checkbox" value="Box 4" required><label>Box 4</label></li>
</ul>

<script type="text/javascript">
$(document).ready(function(){
    var checkboxes = $('.checkboxes');
    checkboxes.change(function(){
        if($('.checkboxes:checked').length>0) {
            checkboxes.removeAttr('required');
        } else {
            checkboxes.attr('required', 'required');
        }
    });
});
</script>