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 a pertinent feat… which you’d have to get through an archetype, because the monk doesn’t have access to it naturally. 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 almost certainly more trouble than it’s worth.
Wearing armor is kind of bad in a different way. 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 4 AC on the ground. For a level 17 monk, that’s 8 AC. 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, but that’s still only up to +4. It’s not inconceivable that you could build a viable monk that ran off of heavier armor, but it would be a very strange build indeed… especially since Mountain Stance already gives you a way to get a decent AC on low-dex monks. Sure, it’s a stance, which means that you can’t use other stances… but wearing armor prevents you from using other stances anyway.
In general, PF2 won’t prevent you from doing any of these things, or even make them necessarily nonviable… but that doesn’t make them good options.