How to toggle a CSS3 checkbox with jQuery?

I've been looking at this demo of a CSS3 checkbox:

http://ift.tt/OByXbA

It is possible to toggle this checkbox (the one in the snippet below) with jQuery?

And, additionally, can I retrieve the checkbox state via jQuery?

I've been trying to follow what happens in Firebug when you click on the checkbox and it seems this is applied:

.squaredThree input[type="checkbox"]:checked + label::after {
  opacity: 1;
}

I've tried the following without any success:

$("#squaredThree").prop( "checked",true);

$(".squaredThree input[type='checkbox'] + label::after").css( "opacity","1");

/* SQUARED THREE */

.squaredThree {
  width: 20px;
  margin: 20px auto;
  position: relative;
}
.squaredThree label {
  cursor: pointer;
  position: absolute;
  width: 20px;
  height: 20px;
  top: 0;
  border-radius: 4px;
  -webkit-box-shadow: inset 0px 1px 1px rgba(0, 0, 0, 0.5), 0px 1px 0px rgba(255, 255, 255, .4);
  -moz-box-shadow: inset 0px 1px 1px rgba(0, 0, 0, 0.5), 0px 1px 0px rgba(255, 255, 255, .4);
  box-shadow: inset 0px 1px 1px rgba(0, 0, 0, 0.5), 0px 1px 0px rgba(255, 255, 255, .4);
  background: -webkit-linear-gradient(top, #222 0%, #45484d 100%);
  background: -moz-linear-gradient(top, #222 0%, #45484d 100%);
  background: -o-linear-gradient(top, #222 0%, #45484d 100%);
  background: -ms-linear-gradient(top, #222 0%, #45484d 100%);
  background: linear-gradient(top, #222 0%, #45484d 100%);
  filter: progid: DXImageTransform.Microsoft.gradient(startColorstr='#222', endColorstr='#45484d', GradientType=0);
}
.squaredThree label:after {
  -ms-filter: "progid:DXImageTransform.Microsoft.Alpha(Opacity=0)";
  filter: alpha(opacity=0);
  opacity: 0;
  content: '';
  position: absolute;
  width: 9px;
  height: 5px;
  background: transparent;
  top: 4px;
  left: 4px;
  border: 3px solid #fcfff4;
  border-top: none;
  border-right: none;
  -webkit-transform: rotate(-45deg);
  -moz-transform: rotate(-45deg);
  -o-transform: rotate(-45deg);
  -ms-transform: rotate(-45deg);
  transform: rotate(-45deg);
}
.squaredThree label:hover::after {
  -ms-filter: "progid:DXImageTransform.Microsoft.Alpha(Opacity=30)";
  filter: alpha(opacity=30);
  opacity: 0.3;
}
.squaredThree input[type=checkbox]:checked + label:after {
  -ms-filter: "progid:DXImageTransform.Microsoft.Alpha(Opacity=100)";
  filter: alpha(opacity=100);
  opacity: 1;
}

<!-- Squared THREE -->
<div class="squaredThree">
  <input type="checkbox" value="None" id="squaredThree" name="check" />
  <label for="squaredThree"></label>
</div>