Will AMR stock rise if Jetblue buys them out?

It wouldn't be Jetblue. It would be US Airways and no it wouldn't help the equity at all.

The way this would go is that US Airways would contact the creditors group and try to work out a deal with them instead of with AMR management. If AMR was not in bankruptcy, US Airways couldn't really take them over without management approval. In bankruptcy, the creditors have a very large say in what happens. Creditor approval of a US Airways takeover proposal would go a long way toward getting a judge to approve the takeover.

Unfortunately, the proposal would be something like US Airways gives cash and US Airways stock to the creditors with value of (say) 40 cents on the dollar (unsecured AMR debt is trading at about 15 right now). The creditors would be delighted. The equity would be wiped out. US Airways would have no interest in talking to anyone who represented the equity interests.

Everyone who thinks that they want to buy AMR stock should instead consider buying AMR debt. For the stock to be worth anything, the unsecured debt needs to be paid off in full. The debt is currently selling at 15. If the stock is worth anything, the debt will go up by a factor of 7. Does anybody out there think that AMR debt is mispriced by a factor of 7?

Edit: I personally think that judging from comments published by people close to the case, that 15 is too much to pay for unsecured AMR debt. My opinion of it is dropping like a rock. A few days ago, I might have been interested in it at a much higher price, but now it's all just gloom and doom.

The company is in bankruptcy. Only a judge could approve a sale and JetBlue doesn't even begin to have the financial means to buy them.

The common stock of AMR is worthless. Don't touch it. It will be canceled and new shares issued after the company emerges from bk.

No body would take over a bankruptcy protected company. Why? Because the market capitalization of the company worth more than the equity. However, other interesting company would take it over after the target company is bankrupt. Because the company now has no good will and the equity worth next to nothing.

It's not likely that the stock holders will receive any equity. The money paid would go to the creditors. Investors get nothing (as it should be).