Is it legal for an employer to fire you for updating you SSN?

In the United States an individual can be fired for no reason because we are nation of at-will employment.  Indeed, there are instances where this can be challenged, but they are few and extremely difficult to contest in a court.  So for a DREAMer who started working with a false Social Security number and then tried to change SS# after being approved for DACA, unless the employer is accommodating and understands your situation, what you have done is to demonstrate to your employer that you lied when you started to work and provided forged documents (so it is not the employer who did anything illegal but it is you who broke the law by working illegally  - do not even think about trying to sue your employer because you don't have a case).  Do not forget that it is a crime for an employer to hire illegal workers so your employer is obviously concerned about charges of breaking the law, but more importantly, an employee who lied and submitted fake papers is generally not welcome at most decent workplaces.  If you cannot be trusted to tell the truth and not break the law, how can you be trusted with being a good employee. 

So instead of putting your employer in an uncomfortable situation of having to fire you, and thus, exposing your criminal actions (definitely this employer will not say a single good thing about you as a reference), if you are an undocumented alien who now is legally authorized to work, simply leave the job on good terms, and find another employment.  Do not even try to use an excuse of stolen identity because no decent company will accept that reason unless it can be documented.