With text boxes it’s not possible, but you can use frames instead: Insert → Frame → Frame will give you a sort of “floating small page within a page”. There are frame styles where you can define colour background, line border, etc, and inside those frames you can use everything Writer has to offer, like paragraph styles.
Another option is to formate one “note” (either a text box or a frame) according to your needs, select it and turn it into an AutoText (Tools → AutoText). Later you’ll be able to insert a copy of that autotext by just issuing its “shortcut” and then pressing
EDIT: To be able to use autotexts with text boxes or frames you need to work a little bit.
On the document insert three empty lines. Anchor the text box to the second one and then select all three lines either with the mouse or with
↑. Now the autotext should work. The idea to make autotext work with objects is to select them “as text”, picking at least one empty line above and bellow their anchoring point (it’s possible to select just the anchor, but not easy, that’s why I suggest to pick more empty lines).