Trying to dig up the origins of ‘don’t have a cow’ and coming up blank. Anyone able to assist? 
I associate it with The Simpsons (Ep. 4 “There’s No Disgrace Like Home”). Both this animal idiom site and this article seem to agree with me, but that’s hardly conclusive…but at least it’s a start?
The Simpsons? No way. My step-sister used that phrase in the mid-70’s.
Haj
I believe the Simspons made the phrase more mainstream, but I’m certain it predates the cartoon by quite a bit.
The Simpsons certainly popularized it, but it definitely existed long before that, though usually used in a descriptive rather than imperative sense - “When your mother finds out, she’s going to have a cow”. Another phrase with roughly the same meaning that I’ve heard is to “crap chickens”.
And I knew it in grade school in the '60s.
An alternative expression is “having kittens”. In both formulations I think it’s referring to the intense-yet-frazzled state of mind of momcats and momcows on the verge of giving birth.
(Actually I’ve never been around momcows on the verge of giving birth but I’ve been around momcats. Seems to me like “don’t have puppies” would be a better phrase than either of them, but momcats can get fussy and obsessive-compulsive)