You can change your snippet to listen to checkbox change instead of removing/adding classes to perform less jQuery operations.
$("#checkbox").change(function() {
$("#checkbox-show").toggle();
});
The HTML markup would be something like this:
<div class="btn-group" data-toggle="buttons">
<label class="btn thisBtn">
<input type="checkbox" id="checkbox"> This
</label>
<label class="btn thatBtn">
<input type="checkbox"> That
</label>
</div>
<div id="checkbox-show" style="display:none;">On checkbox select</div>
If you are displaying the form by using templating engine, you can set #checkbox-show to have display:block on load, only when #checkbox is checked. If not, just perform simple if statement to show/hide div with jQuery only once, on document ready.