You are required to use whatever type of wiring your local code permits for this type of connection. If you are in a residential location then it is likely NMD90; if you are in a commercial location then it will likely require BX/AC. The shielding you are referring to between the can and the box isn't really BX/AC; its just flexible metal conduit. It looks similar to BX but it's not the same thing.
If you do use BX/AC, make sure to use plastic bushings at both ends of the cable so you don't nick the conductors with the sharp BX/AC.
Usually with cans, there is a friction cable clamp you just push the NMD90 through, however if you use BX/AC then you will need to use one of the punch-outs and a proper cable clamp to secure the cable.
Here's what doesn't apply to you, but does apply to most consumer LEDs (sold as primary lighting devices, i.e. screw in bulbs, fluorescent replacements etc):
They are significantly overdriven from spec, for instance an LED will be sold as a 10 watt emitter, however the datasheet will spec it for baseline performance at 1050ma at around 3.3 volts - now hold on, that's only 3.6 watts, not 10! As you read on, you discover they achieve the "10 watt" nameplate by driving it at 2800ma at 3.6 volts, which is allowed, but with lower efficiency and higher junction temperatures requiring bigger heatsinks. Of course that overdrive figures into the LED's life, as the higher junction temperature is one factor in lamp aging.
The other factor is the driver circuit which converts 120/230VAC into the low-voltage, constant-current the LED emitter requires. For a packaged consumer product, this driver circuit is the weak link in the chain, and is the likely cause of death, especially since there are so many ways for bottom shelf brands (FE, UT, LoA) to cut corners here.
In that light, here's what does apply to your LED strip accent lighting.
Those strips usually run on 12VDC or sometimes 24VDC, but they don't include that complex driver circuit. They use a simple, bulletproof resistor for current limiting. Those strips don't have cooling fins, so they don't overdrive the LEDs much. The resistor is also sized for worst-case 14V (or 28V), which is what a car's alternator is putting out to charge the battery. At actual 12V they derate considerably - I've seen a 16 foot "24 watt" strip drop to 15 watts actual. These factors combine to make them dimmer than they could be, but conversely, more long-lasting.
Generally, DC LED strip installations have an AC power supply that is a separate item. It is a failure point, but you can replace it separately.
The other thing is, people really have no idea how long LEDs will actually last. The numbers they throw down are guesses. Realistically they will probably last a lot longer than claims made, but the claims are tempered to avoid appearing unrealistic. My rule of thumb is your installation won't live long enough to see the LED fail - you'll modernize, sell the home or die!
The upshot is, you're already in pretty good shape, and so dimming will help, but not a lot.
I would say the bigger reason to dim it is that lights bright enough to be useful workspace lighting would be annoyingly bright as ambience lights.
and join them in a junction box with a line coming from the switch? I would like to use four of these lights with LED bulbs, so the load will be tiny. It seems like it "should" work, but that's what everyone says just before they burn their house down. I don't know if it makes a difference, but the plugs have fuses. What do you think?
Best Answer
You need to learn the marvelous world of low voltage LED strips
Seriously. You get a common 12 volt UL-listed power supply, and a small junction box to splice, and the world is your oyster.
The strips price out at about $7 for 16 feet no kidding. Not $7/foot, more like 43 cents a foot.
Since you're working entirely in low voltage, you're at basically nil risk of shocking anyone or setting anything on fire. The installation rules are also VERY relaxed, and there's no problem doing what you want the way you plan to.
Which you cannot do with 120V stuff, by the way.