This is not specific to any disease, but indicates that something is either directly upsetting his stomach, or generally making him feel nauseous.
The best thing to do now is withhold food. Make sure he has plenty of water though. When he gets home from the plane ride you can feed him a little (start with two tablespoons) boiled chicken and cooked rice. If he keeps that down after two hours, then you can feed a little more. Gradually introduce his normal food into his meals when the vomiting and diarrhea have stopped.
You may also consider giving Imodium to Jax, so he doesn't soil his cage too much on the journey. Only give it if he stops vomiting though, because otherwise it won't be effective. Below is some information about this drug and dosages.
https://www.petcoach.co/article/loperamide-imodium/For more information about vomiting and diarrhea, I recommend you read the article at the link below.
https://www.petcoach.co/article/diarrhea-vomiting-in-dogs-when-to-call-your-veterinarian/If Jax continues to have vomiting and diarrhea after he gets home, then it would be best to take him to a vet for an evaluation. If Jax has any of the following, then I would consider postponing the flight, because it may be too risky:
1) Blood in diarrhea
2) Very watery diarrhea
3) Starts acting very depressed
4) Stops eating or drinking for longer than 24 hours.
I hope that helps and Jax gets better soon.