Do you need to kill/capture a monster to keep broken parts you loot from the ground?

They usually end up in your field bag or temporary inventory, something like that, but rest assured you'll still get them

They go into your 'quest' item pouch, which is automatically emptied into your box once you complete the quest. If you pick something up, you can check it by selecting 'Item Pouch' in the menu to look at what you got; or sort things and discard something if your inventory is full (or go to camp and store things).

The shiny's? They're placed in your pouch (a separate pouch is used for non-consumables like monster parts) and then sent to your item box after completing the quest. Breaks themselves should show up at the bottom of the rewards listed telling you what part you broke. Just remember that shiny's and breaks are entirely different.

Shinies aren’t breaks. They often drop when breaks happen, but they also happen when you flinch the monster or knock it out. The shinies you pick up are added to your box there and then, they aren’t shown in the rewards screen.

When you break a monster’s part, you get a chance to get an item from the reward pool for that part. If you break a Rathalos wing, you could get a wing or a wingtalon or some webbing or maybe nothing at all.

Thanks all, glad to know I haven't been wasting my time collecting them.

Picking them up instantly adds them, you won't see them on the quest screen.

Items in the "break" section on the quest screen are rare drops you got for breaking a certain part on the monster-- not necessarily the chunk that broke off that you pick up.