At a first glance it sounds like he’s extremely emotional and panicked at the moment, which is absolutely the worst state to be in when it comes to making any long term life decisions.
So what he should do is probably take whatever vacation days he can, go visit his parents and get everything, including himself, in order before returning home to you. He should absolutely not make any long term life decisions in his current state, including something utterly moronic like that resignation letter. It’s quite clear that money is going to be an issue not just for you, but also for the treatment of his mother, so how could he even consider giving up his job like that? Worst case he won’t even be able to help pay for the treatment of his mother. In such a position do not resign from a job before you have secured another! I’d advice trying an argument like that in order to reason him out of that resignation letter. He needs to slow down and make some proper plans before he drives his entire family, parents included, off a financial cliff.
The biggest mistake was probably to fully commit and move the entire family over before the end of your trial period. I’d advice broadening your search to countries that some others already mentioned, like Belgium and Luxembourg since your French might be useful there, and The Netherlands could work with your English. Make sure your spouse is doing the same in order to increase the odds of success. Of course that is assuming that you’re even interested in any of those countries, else going back to the US might be the better alternative.