It depends so much on the person, but in the end, you love them, just as they are. If there is any way for the two of you to develop a sense of humor about his negativity, though, I would try that. This has to be a very gentle and affectionate humor, though. It has to be part of real support, and not just making light of his perception of his problems. It isn't laughing at him; it is an attempt to bring perspective back into play.
For instance, if someone at our house were to say, " "I have to go in early again tomorrow for the stupid project they have me working on that will probably get canceled anyway", then another of us would probably say, "That's the spirit. They can just keep their @#%%ed jack!" This is a code reminder for the following joke:
There was once a man whose car had a flat, way out in the country. He had a spare tire, but his jack broke. At first, he was dejected, because he was miles from anywhere. Then he remembered that he had passed a farmhouse not a mile earlier. He began to walk to the farmhouse, to borrow a jack. As he made the walk, though, he began to worry about what he would do if the farmer would not help him. The more he thought about it, the more certain he became that the farmer would surely not help him. By the time the farmer answered the knock at his front door, the man had himself in a tizzy, and exploded at the bewildered man: "Fine then! Be that way! See if I care! You can just keep your @#%%ed jack!"
We have all sorts of punch-line codes like that. Another one is to repeat this silly rule that we so often find ourselves following: "In any crisis, the first and most important order of business is to properly assess blame." It is stupid to do it, but we all do, and the best way out of it is to laugh at ourselves. As St. Thomas Moore put it: "The Devil, the proud spirit, cannot endure to be mocked."
Of course, after that, if they don't want to laugh you invite them to talk about why they expect such an awful outcome, and look at whether the worst case scenario is really the worst case. In any event, though, it is a gradual play-by-ear sort of process. You learn not what makes them feel better--it is really not comforting for someone to be in a hurry to "fix" you--but what will make them feel loved, for better or worse. If you can express some affection for the fact that they do this to themselves, that can actually help. Mostly, though, you express that they are loved, even when they are being maddening. That is the most important thing. Feeling accepted and loved is a great antidote to depression.
I have had a lot of "negative" friends, and they actually enjoy very much that I have a fond sense of humor about their negativity. You can choose to look at these guys like Eeyore. Their negativity can be so predictable and inappropriate that it can be maddening...but it can also be kind of funny, if it is allowed to be. Eeyores are always going to be kind of like that, but it can become part of what makes them lovable. If you think he might like it, you could even get him and Eeyore coffee mug that says "We love you just the way you are."
This is the other thing that has helped me when, for other reasons, I have wanted to tear my hair out with my husband. I pray: "Lord, you love him, because I can't do it. I just do not have the energy." I have always gotten some help, if only in the form of a softened heart....because when he's hard to love, I'm also stuck in loving, so we both need the mercy of God. If you welcome it, the mercy of God feels pretty wonderful.
Another friend of ours had a good stock agreement for someone who was complaining about a bad situation: "Well, h*&l, d&%n, and spit. That does stink." It just says, "Yep. Bad situation. Those happen. Don't you hate that?" It says that it is OK to not be positive all the time, and that is kind of nice, too.