Froggygirl, it doesn't have to be an either/or situation. There's a third, middle road. You could decide what's best for you, communicate that calmly to your boyfriend and then let him make up his own mind what he wants to do about it. That means not telling him what to do or "encouraging" him to do things your way. As difficult as it is, it's important to step back to let him "man up" and decide what he is and isn't willing to do for himself, you and your relationship.
And you will be doing the same thing yourself. You will be womaning up, which sounds funny. You'll be acting as a mature free agent, leaving the door open for your boyfriend to do the same, if that's what he chooses.
If you want to stay East and if you also want to be with him, you can tell him that and say you hope he will keep trying to find a way to join you.
Then stop giving him the kind of encouragement he finds stressful. He's not an idiot, he knows he has to find a job to be with you -- no matter where you are -- and it is really really, did I say REALLY -- hard to find a job in many parts of the country. You can offer him sympathy and support and love, but stop the pep talks if he's telling you outright that they stress him out.
Five job rejections hurt, it hurts bad, but it's not really all that many in the current economy. If this man would break up with you or demand you come back to a black hole of hopelessness and no opportunities because he's so sensitive he's not willing to take some rejection to improve his life and get to you, well ... he's going to just keep dragging you down if you stay with him.
Please think this over carefully and decide what YOU want to do. Then let him decide if he's willing to bear some inconvenience and pain in order to be with you and to improve your lives.
It's possible that if you step back and stop trying to get him to do what you want, ie, find a job and come East, that you may decide you really miss him and want to go home. Or not. Right now, there is no right or wrong decision when it comes to staying or going home. It's the process of how you decide and stepping back and letting him decide what he wants to do and letting him take his own action that is going to determine how your relationship turns out. Doing that is a lot harder than it sounds. If you can do it, you'll both discover what you're made of.
I wish you the best of luck with this difficult decision-making. I hope things with your sister have improved or will soon improve so that you can make your decisions based on your own needs. You're young. This may be one of the few times in your life when you can make decisions based on your own needs. Once you're married or in a career or if you have children, the freedom to do what YOU want will be lessened greatly.
The decisions you make now, early in your adulthood, may determine the course of the rest of your life. Choose wisely.