Hi Clinton,
It works for me.
Shortcoder does not manipulate the content. What you save is what you get.
I modified the code you shared a bit. (used double quotes, and used https) and it works for me.
<script type="text/javascript" src="https://utah.gov/pmn/meetingsJS.html?entityIds=227&publicBodyIds=&keywords=&widgetWidth=400&widgetHeight=300&borderWidth=1&listSize=50&meetingType=upcoming"></script>