I'm using a custom jQuery 1.10.3 theme. I downloaded every straight from the theme roller and I have intentionally not changed anything.

I created a dialog box and I get an empty gray square where the close icon should be: Screen shot of missing close icon

I compared the code that is generated on my page:

<!-- language: html -->
<div class="ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix">
    <spanid="ui-id-2" class="ui-dialog-title">Title</span>
    <button class="ui-dialog-titlebar-close"></button>
</div>

To the code generated on the Dialog Demo page:

<!-- language: html -->
<div class="ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix">
    <span id="ui-id-1" class="ui-dialog-title">Basic dialog</span>
    <button class="ui-button ui-widget ui-state-default ui-corner-all ui-button-icon-only ui-dialog-titlebar-close" role="button" aria-disabled="false" title="close">
        <span class="ui-button-icon-primary ui-icon ui-icon-closethick"></span>
        <span class="ui-button-text">close</span>
    </button>
</div>

EDIT

The different parts of the code are generated by jQueryUI, not me so I can't just add the span tags without editing the jqueryui js file which seems like a bad/unnecessary choice to achieve normal functionality.

Here is the JavaScript used that generates that part of the code:

<!-- language: js -->
this.element.dialog({
    appendTo: "#summary_container",
    title: this.title(),
    closeText: "Close",
    width: this.width,
    position: {
        my: "center top",
        at: ("center top+"+(window.innerHeight*.1)),
        collision: "none"
    },
    modal: false,
    resizable: false,
    draggable: false,
    show: "fold",
    hide: "fold",
    close: function(){
        if(KOVM.areaSummary.isVisible()){
            KOVM.areaSummary.isVisible(false);
        }
    }
});

I'm at a loss and need help.

This appears to be a bug in the way jQuery ships. You can fix it manually with some dom manipulation on the Dialog Open event:

<!-- language: lang-js -->
$("#selector").dialog({
    open: function() {
        $(this).closest(".ui-dialog")
        .find(".ui-dialog-titlebar-close")
        .removeClass("ui-dialog-titlebar-close")
        .html("<span class='ui-button-icon-primary ui-icon ui-icon-closethick'></span>");
    }
});