This phrase is asking the listener to take action in the positive to help our neighbors.
- 1.a) "Should we not stand by our neighbors who seek to better their conditions in Kansas and Nebraska?"
Switching should to a different place now asks the listener to NOT help out.
- 2.a) "We should not stand by our neighbors who seek to better their conditions in Kansas and Nebraska?"
How is should and not working in this sentence to change the request of the speaker?
First, let me make a slight alteration to your examples, so as to prevent a somewhat possible alternate interpretation (such as ""Should we (pause) [not stand by our neighbors] who seek to better their conditions in Kansas and Nebraska?"). And I think these following are the meanings that you intended anyway, and if worded in this following way, the interpretations will become completely unambiguous:
1.b) "Shouldn't we stand by our neighbors who seek to better their conditions in Kansas and Nebraska?" - - [interrogative question]
2.b) "We shouldn't stand by our neighbors who seek to better their conditions in Kansas and Nebraska?" - - [declarative/echo question]
Version #1.b -- interrogative question: This is a closed interrogative clause, which is typically used to pose a question. (Note that this question is not "asking the listener to take action"--rather, it is merely asking a question.)
Now this question can be asked with either "Shouldn't we stand by . . ." (biased question) or "Should we stand by . . ." (neutral question), and there are possible differences between them, but basically they would both be asking the same question. (As to why or when a speaker might prefer to choose one version over the other, that is its own issue in itself--CGEL "Bias questions" pages 879-86. If you want that issue discussed, then let me know. I'm assuming that that issue is not one that you're interested in right now.)
So, this (a closed interrogative clause) is a typical type of question, one that expects a "yes" or a "no" as an answer.
Version #2.b -- declarative/echo question: This is a declarative clause which has the force of a question. It is used in certain types of context, such as one where the speaker is asking for confirmation of what was just said by a previous speaker (CGEL "echo questions" pages 886-91), or such as one where a question is being asked via the form of a declarative clause (CGEL "declarative questions" pages 881-3).
Here is an example of an echo question:
speaker A: "Our neighbors should stand on their own two feet without any help from us."
speaker B, in response: "So we shouldn't stand by our neighbors who seek to better their conditions in Kansas and Nebraska?" [2.b]
CGEL on page 886:
4.8 Echo questions
The prototypical use of the echo question is to question whether one has correctly heard what the previous speaker said -- heard the stimulus, as we call it. My doubt as to whether I heard the stimulus correctly may arise because it was not perceptually clear (I may have had difficulty making it out above some background noise) or because its content is surprising or remarkable in such a way that I want to verify whether you did in fact say, or mean to say, what I apparently heard.
And now, here is an example of a declarative question:
- CGEL page 868, [5]: You're ready? - - [corresponds to "Are you ready?"]
CGEL on page 881:
4.7.2 Declarative questions
Positive declarative questions have an epistemic bias towards a positive answer, negative ones towards a negative answer:
[38]
- a. They've finished?
- b. They haven't finished?
The expected answer is here the statement with the same propositional content as the question -- i.e. They've finished and They haven't finished respectively. In asking a declarative question I am typically seeking confirmation of a proposition that I am inclined, with varying degrees of strength, to believe. There may be deontic or desiderative bias as well as epistemic, but this is not inherent to the construction as such.
Note that CGEL is the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language (CGEL).