I'm certainly not defending any of the actions of your former employer, but it's v. common for healthcare employers to have a policy of tagging anyone who resigns during orientation as "do not rehire," regardless of whether you gave notice or not. However, plenty of other people have been in your situation and it didn't destroy their careers. Pick yourself up and start thinking about, and practicing, how to talk about this situation in future interviews without
badmouthing the (previous) employer. "It wasn't a good fit" is the standard cliche' in nursing. Resist the temptation, in future interviews, however tempting it may be, to start explaining about how badly you were treated and what a poor work environment it was.
As for whether or not to include that position on your resume' and future applications, did you work there long enough to get at least one paycheck? It's up to you whether you make potential employers aware of your time with them, but, if you got paychecks from them, that information could turn up on a background or credit check (and growing numbers of healthcare employers are now doing credit checks as part of the hiring process). Also, nursing tends to be a small "club" and managers at facilities/organizations in the same community tend to be in communication with each other. If an application specifically asks for all
your previous employers (or all your previous employers in the last X
years), you omit this employer, and someone in nursing or HR at a potential employer happens to find out, one way or another, many (most?) organizations consider that a level of dishonesty that disqualifies you from employment. It would be a shame to miss out on other, better opportunities for that reason, esp
. when so many other nurses have found themselves, at one time or another, in the same situation you're in now (and survived
Best wishes for your journey!