A violent emotional reaction; a conniption; an over-the-top angry response to a situation. More seismic than simply "he shit his pants." A square shit is painful and difficult, due to the presence of pointed corners at the edges, so to "take a square shit" means one's reaction to a situation or its outcome must be equally painful and likely beyond hope.
"My dad took a square shit when he saw what I did to the front bumper of his car."
"I took such a square shit when I heard I was working for Matt New Year's Eve."