Union Square is far and away one of the top restaurants in the city (three stars from the Times, I believe and the top vote-getter from Zagat's annually -- you'll love it). Danny Meyers (who owns it and Gotham and a few others) is known for restaurants that have unsurpassed food, but also a very friendly atmosphere, with comfortable, but professional service. That said, while you certainly could go casual and not be made to feel uncomfortable, personally I would dress up a bit more; a jacket and tie, perhaps?
"What a story........ everything but the bloodhounds snappin' at her rear end." -- Birdie
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"The Devil Be Hittin' Me" -- Whitney