It is recommended, although not foolproof, that one does not leave the company for at least 6 months after GC approval. However, many new laws such as AC21 make this point moot.
Second, this working for 1 month situation - it primarily affects the green card rather than citizenship. Of course, the IO can check anything.
Last, even though it sounds bad, there is less and less incentive for CIS to go back and scrutinize papers unless they believe they can prove outright fraud. How do you avoid that perception? Hard but the following might help:
A) Do not file until 5 full years have completed since the GC was approved. CIS can discover a GC grant mistake and file to revoke the GC before the 5 year anniversary, but it becomes harder thereafter.
B) Also, why not delay filing until the employer B is completely off the books - that is 5 calendar years for which he may or may not be asked to provide transcripts should be after the calendar year he stopped working for this employer B. Is it really necessary, NO. But if he can not explain why he left the employer besides "another opportunity", this might help just a bit.
None of this is legal advice....