Month and year are fine.
Give all your previous names even if there is no record of them in the U.S.
Andrew M. Bramante, Rosner Partners, 216-771-5588. Free telephone consultation. You should always consult with an experienced immigration attorney to make certain that the advice you received is appropriate for your particular immigration case.
Always be forthcoming, truthful, honest and complete in any forms including N400 and include all information asked for in the form. Advise being represented in any proceeding before USCIS; even if you think your case is not hard or straightforward. Search Avvo, have an attorney represent you in filing and go with you at the time of your interview so find an attorney in your area.
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The most important thing on any application is that you are honest. Dates of employment - they want the complete date, but if you are only able to put in the month and year then do that. As for your previous names - You should put all names you have ever used. That is what the question asks and that is what you should put.