Edit the ‘box’ if it’s a container.
Select all of the ‘box’ geometry [edges and faces], then Edit > Cut.
Edit the ‘tall’ component.
Edit > Paste-In-Place
This inserts the ‘box’ geometry into the tall component [selected].
If you need to, use treble-click or a ‘fence’ to select all of the box’s geometry.
Use Context-menu > Reverse Faces [this is needed later…]
Use Context-menu > Intersect > with Context.
The ‘box’ now cuts the ‘tall’ geometry.
Erase the unwanted outer parts of the ‘box’.
Also erase the unwanted face in the cut mid-portion of the tall part.
You should now have the tall part cut into two pieces, each with a properly oriented face on the newly cut parts.
Note how reversing the box’s faces when I said to, set them up to be suitably oriented in the next step when you tidy up.
You don’t have to do it then, but if you don’t you have to select the incorrectly oriented faces and reverse them later anyway…