A monk can use a shield just fine... it's just that it's not necessarily worth bothering.
There's nothing about FoB that requires you to have two hands free. It's a flourish that gives you two unarmed attacks for one action. That's pretty good, but it's only really notable in how early the monk gets it.
For wearing a shield, as long as you aren't using a tower shield, the only thing it costs you is use of one hand, and the one bulk for lugging the thing around. A buckler costs even less. On the other hand, without pertinent feats, it doesn't really give you all that much either. In order to gain the AC bonus from having a shield you need to take an action in that round to raise the thing, and you can't shield block without burning a general feat on it. That one hand isn't going to cost you all that much under normal circumstances, but it will slow you down if you decide to go in for combat maneuvers in a big way. As soon as you grapple someone with your other hand, you'll suddenly find your options constrained in a way that they wouldn't be without the shield.
Armor is likewise possible... and a poor choice.
Wearing armor, though, is just bad. Specifically, the monk has no native proficiency in any armor, while their unarmored proficiency is quite good. Effectively, as soon as you put on armor as a level one monk, you're dropping 5 AC on the ground - 4 for going from Expert to Untrained, and 1 for the level... and that gets 1 point worse every level, and then worse still when the monk unarmored proficiency goes up. For a level 17 monk, that's 25 AC (as legendary down to untrained is 8). Armor base AC maxxes out at +6, and there's no way to get any armor where AC plus dex cap is higher than +5 without going for heavy armor... or going completely unarmored and having a dex above 20. Also, armor has bulk of its own, along with various other drawbacks. Now, it's possible to get up to expert skill in heavier armors with archetype feats, and that helps out a lot, both by getting you up to +4, and (more importantly) by letting you add your level back in. It's still going to cost you precious class feats and wind up with an AC that's almost certain to be worse, though. If for some reason you wanted to tank your Dex into the floor, it's not inconceivable that you could wind up in a situation where heavy armor plus skill was better than dex plus (higher) skill, but the obvious answer to that would be to instead send those class feats on Mountain Stance, which already gives you a way to get a decent AC on low-dex monks, and still gain all the advantage of the impressively high monk unarmored proficiency. Sure, it's a stance, which means that you can't use other stances... but wearing armor prevents you from using other stances anyway. Putting heavy armor on a monk is basically a mistake, and you almost certainly shouldn't do it.