Quid Est EEK?
They derive from something my fiancee said.
We were looking at pictures of girls hugging and holding each other. She said "EEK! EEK! Lesbians!" because she was parodying the idea that when two female characters are close to each other, there must be something (as in a "more-than-friends" something) between them. "EEKy" and "EEKiness" and "EEK" parody the aforementioned idea. "EEKiness" refers to something... anything... that could be construed as lesbianism.
Edit: Nowadays, "EEK" and its various derivatives often are used when discussing about any sort of female-only relationship that others could interpret to have some kind of implicit or explicit sexual or romantic significance. Relevant relationships could be anything from glances between two female people who do not know each other all to hardcore yuri relationships. In any case, it is used jocularly.
LWotP: tribadismus -i m. "lesbianism" (Vocabulum Milleri?).