Html.CheckBox helper adds a hidden field to the form. I would suggest you assigning an unique identifier to each element in the Children collection and then have this in your form instead of using the helper:
<% foreach (var child in Model.Children) { %>
<input type="checkbox" name="childrenToDelete" value="<%=child.Id%>" />
<%}%>
And then in your controller action:
public ActionResult DeleteChildren(string[] childrenToDelete)
{
// childrenToDelete array will contain selected ids of children to delete
return View();
}