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Sorry, that is linking weird, but you want the part number 2404-03-02-SS, 3/16" x 1/8"-27 NPTF. That 3/16" is a JIC thread, which is why it is confusing. The 3/16" JIC Flare fitting has a 3/8" male thread. It's the same reason that the part he used in the instructable has the thread listed as "3/16 (3/8'24)."
There's a lot of conflicting information on this. Most things I've read say the paintball tank fillers get their gas from the same places as restaurants and bars. To be safe, it's probably best to ask when you're getting it filled, but it really doesn't seem to be an issue for the most part.
Yes, a flared connector works fine for me. This is the one I use: http://www.amazon.com/Brennan-2404-03-02-SS-Stainless-Fitting-03MJ-02MP/dp/B005C7KDJO/