J: Rygel has been possessed by some MENSA-member virus...
MENSA is an organization for people with high IQs.

J: Shall we shaft it?
Giving someone "the shaft" is American slang for harsh or unfair treatment. Definition courtesy of Dictionary.com.

Z: We must formulate some sort of antibody that will kill the virus without killing Rygel.
J: And what if you don’t? You might kill Rygel. You might think your magic pill worked, and then Rygel’s up walking around the ship, coughing up spores, cats and dogs living together.
"Cats and dogs living together is a Ghostbusters misquote. Listen to a wave of this scene from Ghostbusters.

Referring to the Peacekeeper science officer--
J: Look, as soon as I leave here, I’m going straight to hot lips.
Loretta Switt played Margaret "Hot Lips" Houlihan on M.A.S.H.

J: Eenie..meenie...minie...D’Argo.
"Eenie, meenie, minie, moe" is a child's rhyme sometimes used to make a choice. The entire rhyme goes as follows:

Eenie, meenie, minie, moe
Catch a tiger by the toe
If he hollers, let him go
Eenie, meenie, minie, moe

A: Thank you.
J: Don’t mention it.
A: Why would I ever mention it?
"Don't mention it." is an English language idiom sometimes used, when being thanked, as an alternative to the standard response of "You're welcome."