The New Partridge Dictionary of Slang and Unconventional English (2008) says:
have a cow to become emotionally overwrought; to lose control US, 1966.
Speaking of Animals: A Dictionary of Animal Metaphors (1995) by Robert Allen Palmatier says:
HAVE A COW to have a cow. To have an anxiety attack. Source: COW. WNNCD: O.E. On the TV show "The Simpsons," Bart Simpson says "Don't have a cow, man!" meaning "Don't get all upset about it." Bart is likening an anxiety attack to giving birth to a cow - a frightening thought. Normally cows are the ones that give birth to cows - i.e., bull calves and heifer calves. Compare Have Kittens.
WNNCD is Webster's Ninth New Collegiate Dictionary (1983) and O.E. means Old English, but the O.E. must apply to the plain word cow rather than the phrase. (The OED dates cow to Old English.)
This Yahoo Voices article - Idioms Unpacked: "Don't Have a Cow" - also claims it means to (not) give birth to a cow, which would be distressing for a human to do. It lists a number of references at the end, but I've not followed them.
A quick search of Google Books shows this snippet dated 1962 from Field and Stream, Volume 67:
"Oh, don't have a cow," Chip said confidently. "They just haven't begun to fly yet."
"If they don't fly soon," Andy insisted, "they're going to need landing lights."
(Care must be taken with Google Books' snippets as they're often mislabelled, but following the story text we find an advert for a "NEW 1963 book of homes", so it's likely from 1962 or 1963.)
Searching Subzin.com, the first film I found to use the phrase was Sixteen Candles (1984):
00:39:00 I don't know, Jake.
00:39:02 I'm getting strange signals. Well, they're not comin' from me.
00:39:05 Everything's fine. Don't have a cow.
00:39:08 Okay.
00:39:10 Just remember one thing.
Edit: Good timing, as the OED have just released an update to the dictionary containing the phrase for the first time. The first quotation is from a 1959 newspaper:
1959 Denton (Texas) Record-Chron. 26 Mar. 3/2
He won't let me watch rock 'n roll shows... He'd
have a cow if he knew I watched 77 Sunset Strip.
It generally means "it is your responsibility." By responsibility it means that the blame or the victory would belong to you - sometimes it is said when someone is taking a risk or doing something that could go badly. The person saying it means "it is your choice, but don't blame me if it fails." Particularly in superhero movies where people are doing dangerous things all the time.
Best Answer
I would recommend having a look here: I'm Game
And Since its an Idiom there is really no correct way of using it. A few examples:
Its not cool:
Its not worth/fun/cool:
disagreeing:
I think the 2nd example is of relevance to you.