Most recipes say to use a slow cooker, but some say boil. Why would one be better than the other?
It is better to slow simmer. As a very general rule, it is better to moist cook meat or poultry slowly. It makes for a more tender, more luscious bit of meat. Sometimes I'd rather not reinvent the wheel, so I'll just show you this from Bon Appetit to address the question of why.
I plan on using boneless skinless chicken breast.
Certainly that is your prerogative! :) I would advise against it. In my opinion, boneless skinless thighs are best for this application, even though breasts shred more evenly and are just prettier than thighs. Again, I'm going to provide a link, this time for a recipe Add a Pinch . Easy!!
I recognize there is no exact time for it to be done, so what are the signs of being done and ready to shred?
One of the reasons I love chicken thighs is that they are almost infinitely forgiving. You (for a few reasons, safety being perhaps the most important) should always cook chicken to an internal temperature of at least 165F (74C). In my opinion thighs are better much longer cooked than that, but that's the minimum for safety. When in any doubt, use a digital, instant read thermometer. That one and this one tied America's Test Kitchen ratings for best inexpensive models.
Last question, how much water would I use in a slow cooker with the chicken?
None :) Use a very small amount (2 TBS even) of something wet and flavorful! The recipe in the above section should be able to help with that. I use the "refrigerator velcro" (thanks Alton Brown for the great line) method. Open the refrigerator and look for something wet, Mexican-y and isn't spoiled. That'll work!
Good Luck.
I've left a lot of room for improvement in this answer and for more answers. I'm pretty sure the community has quite a bit to add.
Best Answer
Under such tight restrictions, one step that might help is brining. This would apply to chicken breasts, pork loin, or pretty much any other lean meat; darker cuts such as chicken thighs will benefit less, but there's no reason you can't brine those if you want to.
You can find a quick primer about how and why this works here; in short, brining denatures some of the proteins that "squeeze" moisture out of the meat as it cooks, which reduces moisture loss in the finished product. You don't really need anything other than salt, water, and time, though many recipes will call for additional flavoring agents.
Make sure that you don't overcook to begin with. If you can, get a thermometer and measure the meat's internal temperature. Here is a decent guide to the recommended temperatures for various types of meat; remember that you want to cook to slightly below these temperatures to allow for carryover.
From personal experience I would advise you to vary your technique and try to cook different variations on the same basic food; this guards against boredom, and makes it easier to stick to a regimen of weekly meal prep. Instead of broiling, cook your chicken on the stove, then make a basic pan sauce to serve over it, which will add moisture and flavor. If chicken thighs are on sale, get some and braise them instead (you don't need to bother with brining if you do this). This takes some time to cook, but it's very easy, and they'll reheat well for days.