Any legal remedy when quitting old job for a new one that was later cancelled?
If you quit an existing job for a new one that was cancelled, you very rarely have any legal remedy unless you had a written employment contract guarantying you the new job for some fixed period of time. Otherwise, "employment at will" means you were not guaranteed a job.
"Better the devil you know than the one you don't"--that's an old saying that has a great deal of currency in the employment context.
Your current job may have its flaws, and a promised job may offer benefits or opportunities you don’t currently have, but you know your current job and employer. You don't know the new employer, or what they will do or how they will act. This includes not knowing if they will keep their promise of employment or renege on it--either terminating you shortly after you start, or even cancelling the promised job entirely.
Employment in this country is "employment at will" except when you have an enforceable written employment contract. the employer can terminate employment at will without penalty, reason, or prior notice, regardless of the effect on the employee--which includes being able to terminate employment before it even begins (i.e., cancelling a promised job). Quite simply, without a written employment contact, you have no right to or guaranty of a job.
If you do have a written employment contract, you can enforce its terms. If the employer fired you--or didn't even let you start work!--in violation of a contract, you could sue for monetary compensation (e.g., for the salary you were contractually guaranteed). To be an enforceable contact in this context, it must be in writing and it must be for a definite term, such as a one-year contract. A seeming "contract" which does not restrict the employer's ability to terminate you for some fixed period of time does not restrict or restrain their ability to terminate your employment, or to take away a job before you even start.
The said, there is a legal doctrine that sometimes can enforce a non-contractual promise. Called "promissory estoppel," it prevents ("estops") someone from denying or not following through on their promise when ALL the following criteria are met:
1) someone made you a promise (i.e., a job) to do something (I.e., take the promised job);
2) to do that thing, you'd have to do something to your detriment (i.e., quitting an existing job);
3) the person making you the promise knew you'd have to do that detrimental thing (e.g., knew you were currently employed, and would have to quit to take the offered job), but nonetheless, in full knowledge of that, made the promise anyway;
4) it was reasonable for you to rely on the promise; and
5) in reasonable reliance on their promise, to did that thing to your detriment (e.g. ,quit your old job).
Criteria or factor 4), above, is apt to be the sticking point in this context. Given that employment in this country is "employment at will," it can very hard to convince a court that it was reasonable to rely on a promise of a job when there was no contract.
Employment at will makes job promises inherently unreliable when there is no written contract backing them up. To prevail under this theory, you'd have to show that despite employment at will, there was some reason why it was reasonable to rely on this promise of a job. Quite frankly, the odds are against succeeding, since holding an employer to a non-contractual promise undermines the prevailing policy of employment at will. However, if there were circumstances or communications making this particular job offer especially reliable—seeming that you may be able to hold them to their promise of a job--the doctrine of promissory estoppel offers at least the possibility of enforcing their promise. This is something you should discuss with an employment law attorney if you feel that the criteria for applying it have all been met.