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Squee has had wide-spread usage in fandom as both verb and noun, as an onomatopoetic expression of enthusiasm and joy. The word sums up fans' feels of wholeheartedly embracing one's favorite story, or character, or fannish moment.
A squee represents a sound which is a combination of squeal, squeak, and scream.
In general, a common feature of (self-proclaimed) fangirling is to focus primarily on squeeing for one's BSO rather than on analysis, criticism, or topics which might be less amenable to dolphin noises. "I fangirl that show so hard" is a common format for expressing squee, with the understanding implicit that the speaker does not want to be required to explain or justify her reaction.
Some fans think "squee" is an antonym for squick, as both words refer to a visceral, sometimes unreasonable, set of feelings. Finding a fanwork that expresses one's kink might cause one to squee, but kink and squee are not synonyms. Rather, the kink is the wish, and the squee is the sensation of having one's wish fulfilled or the enthusiastic response it engenders.
"Squee was first used as comic book sound-effect word made by dying robots." One example is "Magnus Robot Fighter 4000 AD," a fictional comic book superhero created by writer/artist Russ Manning in 1963. 
In Professional Wrestling, the trope makers and codifiers are The Rock N Roll Express, who were created as a B Show replacement team for The Fabulous Ones with the difference being that the Rock N Rolls were marketed toward teen girls with a safe Tiger Beat image, as opposed to how the Fabs were marketed toward adult women as Playgirl-style sex symbols. The high-pitched reactions Jeff Hardy would receive in World Wrestling Entertainment starting in 1999 are directly in the tradition of the RNRs.
In September 2016, squee was added to the Oxford English Dictionary.
By 2016, as fandom moved to Tumblr, the word has fallen out of favor.
Usage and Examples
- "(I can't believe I'm adopting this word, sigh)--"squee" (sigh) of "I'm in love with a writer" and I don't know where else to post about it." 
- "Fangirliensis: Infatuation with a canon/character/canon creator to the point of fainting and squealing whenever it is mentioned. Common Symptoms: Inability to spell. Use of netspeak or "creative" misspellings, such as "hawt" for "hot." Use of the word "Squee!" and many exclamation points. Inability to tolerate criticism of the object of fangirling." 
- "I squeed like a 14-year-old fangirl when I found out."
Harshing the squee
"(to) harsh someone's squee" (also known as "pissing in [someone]'s cheerios") is to respond with criticism to an expression of squee, or to criticize something which another fan is squeeing over, and thus spoil their enjoyment of it. Harshing someone's squee is sometimes considered an unkind or uncouth thing to do; in other instances it is an unkind term applied to a simple difference of opinion. Some fans will begin their squeeful reactions with a statement intended to ward off well-meaning people with less positive reactions; others routinely warn that they are not entirely squeeful, and would rather not receive Pollyanna or enthusiastic responses.
- The true origin of "squee!", Archived version, January 1, 2014
- squee, int. and n. : Oxford English Dictionary, Archived version
- New words list September 2016, Archived version. New words notes September 2016, Archived version
- Speranza at Reporting in from Racine Street (July 25, 2002)
- from Diseases of the Common Fanfic Writer, V. 1.0 by Limyaael (October 29, 2003)
- In August 2013, squee was added to the Oxford Dictionaries Online. Also, Buzzworthy words added to Oxford Dictionaries Online – squee! Oxford Dictionaries Online blog, accessed 2013-9-12
- Zen and the Art of Not Pissing in People's Cheerios, livejournal post by cereta March 23, 2007.