Elizabeth said:
Beikoku -- America, though, not the US
Hmm? I might be misunderstanding you, but 米国（べいこく）
is the United States. It's just a formal/written-style equivalent of アメリカ, with both of them being abbreviations of the official name, 亜米利加(アメリカ)合衆国. (BTW, for anyone else -- Elizabeth and Paul already know, I'm sure -- who is curious why the US is 'rice country', you can see that it just comes from that second character above in the phonetic kanji representation of 'America'. The reason they don't use the 亜 is because it's already the abbreviation for 亜細亜（アジア）)
Anyway, you can get 北米（ほくべい） and 南米（なんべい） for the continents of North and South America, but 米国 by itself is simply referring to the good old US of A. Check the definition from 大辞林: 【
米国】 (Actually, on the other hand it seems that 【
アメリカ】 can apparently be used as a cover term for the アメリカ大陸, although I would imagine that it's much more commonly used to refer to the USA.)
some examples:
米国大統領選（べいこくだいとうりょうせん） = the U.S. presidential election
日米関係（にちべいかんけい） = Japan-U.S. relations
...neither of which have anything to do with any part of the American continent other than the United States.
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Also, backing up to a previous question.
Elizabeth said:
誰でも sounds more like the proper subject of a sentence than 誰にでも, but that's just a vague impression until I get more information.
I can understand not quite being able to distinguish between, for example, 「誰にでも落ち込む時はある」 and 「誰でも落ち込む時はある」 (I'm not sure myself that there's a _significant_ difference in nuance here, although there are certainly other sentences one could think up in which one or the other would be inappropriate), but I'm not quite sure why you're having a hard time accepting 「誰にでも」 as grammatical.
I mean, it's perfectly possible to see other particles/question words used in a similar construction, like: (quick examples pulled from Google)
誰とでも無理なく話せる
誰とでも仲良くなれる方法
どこへでも気軽に携帯できる
自分が思ったことを
どこにでもいいから、書いておいたほうが良い。
誰からでも学ぶことができる☆
Or is it something else that was bothering you about 「誰にでも」?