Meog ji anhayo: A less formal (but still polite) way of saying you don't eat. Again "meog" is eat, "ji" is a connector used with verbs, "an" (like earlier) makes it negative and the "--yo" makes it polite.
(If your phrasebook has "an meogayo," that's basically the same thing but didn't fit the song rhythm as nicely.)
Bbego juseyo: "Take out please" ("bbae" is the stem of remove, "go" is a connector used with verbs, and "juseyo," literally "give me" makes it nice and polite.)
Gogi: Meat (Tweiji: Pig, Soe: Cow, Dalk: Chicken. Yes you have to say all three. "Meat" by itself generally is taken to mean beef. I've seen ham in "vegetable" kim-bap (sushi-like) rolls. See http://taegukilchang.blogspot.com/2009/07/no-meat-doesnt-mean-no-meat.html.)
Modun chongnyu gogi: ("Modun" means every ("modu" in the adjective form) "chongnyu" is kind, and I think we've covered "gogi" by now. Again for those who can read and are surprised at the spelling: riul ("r") after iung ("ng") becomes "n.")
Haemul: Seafood. (Mulgogi: Fish (which is not seafood sometimes), Seu: shrimp (and lobster but let's not go there), Joge: Shellfish (usually clam). Yes you have to say odang seperately, it's not considered seafood. Again, see http://taegukilchang.blogspot.com/2009/07/no-meat-doesnt-mean-no-meat.html.)
Tarun keot i. First, you probably noticed I say "kosh-i" that's because shiot (romanized s), when in the padchim (bottom) is said like a "t," but when followed by a vowel (like "i," our dear old topic marker), it usually is said as an "s" and part of the next vowel. That said, "tarun" is another and "keot" is thing. Since Korean usually doesn't distinguish between plural, this could be either "another thing ..." or "other things ..."
Kyeran: Egg. In a dictionary there are a ton of words for egg. This one works in restaurants. I have no idea what the rest are.
Uyu: Milk. Kurim: Cream (duh), Aisu Kurim: Ice Cream (duh again)
Meog ji anhaso: Same as "meog ji anhayo," but changing the -yo to -so makes it a reason for doing something else. Here, "because I don't eat these things ,... " (followed by "please take them out," "bbego juseyo").
Hope you enjoyed it. Veggies in Seoul have some resources, like this one: http://seoulveggieclub.wordpress.com/.
The rest of you, though (like us Wonju-ites), best get to singin'