I have no problem getting them to use the better weapons I pass to them. (You can tell they've switched because the Equipped marker moves to the new weapon.) But I've given full suits of armor to my settlers assigned to Security, but they never put any of it on. (No Equipped markers.)

Give them the armor and then hit the T key (pc default) don't know what it is on consoles.

Press (Y) or triangle if you're on console, at the equip screen.

I'm on PC. I take it I have to be in Build mode with the settler selected?

No, you have to talk to them, then do the trade option.

Nope

Just speak with them, open the trade menu, give them what you want, press "T"

So now that I've already given them the armor that they're not wearing, just talk to them and hit T? Or is that only an option only when they recive armor from me? (T-ing after each piece is given?) What happens if they have two different types of armor for the same limb, and I hit T? Do they choose which they prefer? (Better physical Damage rating, or better energy Damage rating? Hmm.)

Does this also work for Companions like Preston Garvey?

Its really easy. Just highlight the piece of armor or weapon you want them to have equiped and press "t". If the item is equiped it will have the little box appear next to it.

THANK YOU!