I suggest you read the transcript to the Good Eats show on freezing, or, better yet, find a copy of the episode to watch. (Good Eats is an American TV show focusing on food science.) Here are some relevant excerpts:
[In the freezer,] any present bacteria are going to have a devil of a time finding the moisture they require to live, breed, survive. Since they themselves contain a considerable dose of H2O, freezing halts bacterial activity altogether, though some of the little beasties can survive to decompose another day. While it is in this frozen state, the meat will remain unspoiled for years. But there's a problem. You see, home freezers work very slowly. That means that as those little H2Os start to line up inside, they'll create huge, sharp, jagged crystals that will rise up, slicing and dicing their way through cell walls, muscle fibers, and pretty much anything else that gets in the way. Now while frozen, you won't notice this damage. But when it comes time to thaw, all of those perforated cells will start to leak out moisture all over the place. It's called drip loss and it's not a sign of good eats.
Therefore, if you are buying the meat fresh (i.e., it is not already frozen), then you need to take special care in how you freeze it:
One good way to prevent drip loss is to freeze the target food very, very quickly to a very, very low temperature. So that instead of creating huge, jagged, nasty ice crystals, you get very, very tiny little ice crystals.
The episode goes on to describe a process for freezing meats at home (basically, the idea is to use small pieces of meat, freeze them separately on a sheet tray, and then once they are frozen transfer them to a bag for long-term storage).
Even though the meat will technically be safe to eat for years, there are a few caveats. For one thing, the environment in the freezer is very dry. That can cause moisture on the surface of the meats to sublimate and form ice crystals, otherwise known as freezer burn. This can be avoided by packaging the meat in a vacuum (e.g., vacuum sealed bags), however, if you don't have a good vacuum sealer, I would suggest limiting their freezing to 6 or 8 months in order to avoid freezer burn. Another reason to make sure you use an air-tight container is that fats are very good at picking up flavors of other items you might have in the freezer.
I am fairly certain the "fats will surely go bad" claim is a myth, especially since fat alone is a good preservative. Perhaps you should ask that question on Skeptics.SE.
Speaking as pure theorist here, because I've never compared the saltiness of cooked chicken.
I think that there are two things you will have to think about. The first one is osmosis (water that gets soaked into the meat through cellular membranes), the second one is transport through porous media - like a sponge, the open ends of muscle fibres of your chicken soak up the liquid you put them in. What you want to do is to prevent osmosis into the chicken (because only pure water will go in through osmosis, concentrating the salt outside) and maximize absorption. I will assume that 0.9% saline is isotonic for a chicken, and I think this is a safe assumption, because veterinary saline isn't marked as species-specific. Seeing that your soup liquid is 0.27% saline, you can expect some osmosis to happen. *
Let's tackle the more interesting part first. You want more absorption to happen. The absorption is described by the Washburn formula,
and you want to maximize your L.
L improves with time. Luckily, a soup is meant to be cooked for a long time anyway. But when you wonder whether to give it some more time on the stove or not, more time is probably better. Keep the temperature lowish for less evaporation, and a lid on which will drip back evaporated water.
L improves with lesser viscosity. No way to influence it in your basic recipe, but in a real soup, avoid thickening. So don't use starchy ingredients, or at least, wash them before adding.
Surface tension should be kept high. Again, this is ingredient-specific. No way to influence it in the basic recipe, but it could turn out that some vegetables are reducing your surface tension a lot - I can't think of a soup vegetable or a herb or spice which is famous for a high saponine content, but it could happen. Also, you should give your pots a finishing wash with clean water if possible - detergent and dishwasher finishing liquid reduce surface tension a lot.
Pore size. My intuition says that this should have the biggest effect in the soup case, as you probably can't influence viscosity and surface tension too much. Obviously, the bigger your holes, the more water comes through. A good way to do that is to thoroughly denature your proteins. The first and most common way for that is prolonged cooking time. Second, brining in concentrated saline (6%) and/or acid before you cook will attack the meat surface, again denaturing proteins. However, I don't know how wise it is to use this option, because some of the brine will get absorbed into the chicken, which will leave less space for broth and will increase salt content as a whole. (This assumes that you salt the broth; Brian's idea of not adding salt to the cooking water after brining has merit). Third, you can use meat from the freezer (this is a very likely reason for the mixed results you saw until now). Freezing produces ice crystals, which rupture cell walls. When you use the thawed meat in a soup, there are more holes for water to flow in.
The Washburn formula is for a single capillary. But the more capillaries you have, the more absorption you get. So, what you want to do is to cut the chicken meat perpendicularly to the muscle fibres. And cut it into many small pieces instead of a few big ones.
Now we took care of the absorption, let's look into the osmosis. You can't change the salt content of the chicken's cytoplasm. But for osmosis, you have to separate the two solutions by a semipermeable membrane (the cell wall). Poke a hole in the wall, and the osmotic gradient vanishes when the liquids mix. So everything from the third point on absorption helps you reduce osmosis too.
As for your suggestions: 1. I see no reason why slow or quick heating will change absorption. I guess the products of a Maillard reaction could clog some pores, if you sear before cooking, but it won't happen during boiling. As I said, cook slow because of evaporation. 2. The chicken shouldn't absorb salt, but salty water, see the footnote. Although, if there is a hole in my theory, this is a likely place for it. 3. I doubt it, but if you wash it, you could free eventual clogged openings. 4. Certainly, cook longer.
To summarize: Cook meat which has been frozen, cut in small pieces perpendicularly to the grain, and stew it for a long time.
* I don't see a mechanism for the chicken absorbing a higher percentage of salt than what you have in the broth. I assume that in your "normal" cases, you are left with salt content near the initial 0.27%, and in bad cases, this gets concentrated.
Update As requested in the comments, I am providing an explanation about "cutting across the fibers". Short story, meat is made up of muscles, and a muscle is made up of fibers, or bundles of bundles of cells. You can easily see them in raw meat. You want to slice across them, so their ends are open, as opposed to along.
The good, long explanation with pictures can be found on Serious Eats
Best Answer
That release of moisture is due to breakage of chicken's cell structure, and further moisturing will not repair it - so it's not about the pores, and there's no "going back" from that state. Still, if You cook Your chicken in a liquid with agents that affect osmotic pressure (salt, for example), it will lead to release of liquid through membranes, and further cooling will cause the reverse effect (chcken absurbing released liquid). That's why stew/soup-like dishes often taste better after cooling/reheating.
Hope I got the point)