Can Bob Swordnboard, the full-plated fighter, "enter the stance employed by the fighting style" if he has a style-feat? Or is being unarmed/unarmoured a requirement?
Yes.
Hold it. That yes is only partially true, the real answer is both yes and no.
All of the styles can be entered (which means taking a swift action to start it up) regardless of currently equopped weapons and armor.
However, most of the styles have special loopholes of their own outside of their prerequisites to get a benefit.
For example, Boar style requires you to hit with two unarmed strikes, while crane style requires at least one open hand. Read the styles carefully to figure out if the style is right for you.
Actually, a remark by Jiggy in this thread had me thinking about a Snake Style Bard, perhaps with a level of Unarmed Fighter to get the style feat. So really all such a PC would want is to enter the style while wearing light armour and shooting a bow or flourishing a rapier.
I didn't see anything in the rules prohibiting it; it just feels weird.
I'm not too familiar with bards, how's their swift action economy?
Most Bard spells are a standard action to cast, and while, come 13th lvl I think, you can start a BP as a swift action, there is nothing saying you couldn't still use a move or standard action to do so either, as is required/permitted as lower levels.
Furthermore, once started it only takes a free action to maintain a BP. And as far as I know there are no other taxes on their swift actions from other class features, so overall their swift action economy is pretty good I'd say.