1- I'm just reciting USA'S neighboring countries.
Correct, except it should be "the USA's", as it is "the United States of America's".
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2- I'm just reciting the neighboring countries to USA.
"the neighboring countries of the USA"
3- I'm just reciting neighboring countries of USA.
This is fine (again, needs "the"), but is subtly different in meaning. If you say "reciting the neighboring countries" then you're saying that you're reciting all of them, whereas "reciting neighboring countries" could be just some of them.
4- I'm just reciting countries neighboring USA.
This is also correct (if used with "the USA"), in the sense of "reciting countries that neighbour the USA". It would be equivalent to sentence 3 in that sense (and sentences 1 and 2 would be equivalent if corrected).