Here in the UK, employees can't work for more than 48 hours per week unless they've signed a written agreement to 'opt out' in advance (although 'domestic workers in private households', which might include nannies, are outside of the regulations).
Further, all employees must have either one full day off per week or two full days off per fortnight. They must also have a minimum of 11 hours rest between completing one days work and starting the next. Assuming that they work either 5 or 6 days per week, they must also be given a minimum of 26 days paid holiday per year. (The employer is free to determine the dates of those holidays but, once agreed with the employee, the employer most definitely can't change the dates at short notice, as has happened in your case).
Workers in some other European countries (such as France) consider the UK's laws to be nowhere near sufficient (with France, for example, having a maximum 35 hour working week, although that's currently under review).
The USA has some of slackest employment laws in the world (with, for example, a far lower paid holiday entitlement than in almost any other western country). Even so, I find it unlikely that an employer can legally force an employee to abandon their holiday plans at the last minute (as, if for no other reason, that would leave him open to litigation in respect of the employee's losses through cancelled holiday plans).
Here in the UK we also have free employment rights advice easily available through a public body (ACAS) and Citizens' Advice Bureaus (staffed by trained volunteers) in every town. However Google seems to suggest that there are no equivalent bodies in the USA.
Your best chance of getting an answer from someone who knows the Federal or State laws that apply where you are is probably to post your question again on this US-based website:
https://answers.yahoo.com/
(I suggest mentioning which state you're in, in case there are different laws in different states).
I apologise for the fact that this post doesn't actually answer your question (simply because we Brits don't understand US legislation) but I hope that it points you in the right direction. (However Gness might have already given you the best advice: Find another job!)