A squee is the sound made by excited fangirls (or fanboys). The exact nature of what makes someone squee depends on the type of fan. South Park fans squee at the announcement of a new episode, and shippers squee when something happens with their favorite pairing.
It is an excited expression of desire for the object that caused the squee, though not necessarily sexual desire.
In writing, it is often followed immediately by the emote *dies*, to imply that the user in question has suddenly fainted from euphoric overload.
Most likely a mutation of or derivation from "squeal". Possibly a Portmanteau of "squeal" and "glee". Any resemblance to Squick is (mostly) purely coincidental. Although, it is important to note that one person's Squee can be another person's Squick.