Screw only needs to be green if the screw itself is the MBJ. Otherwise the MBJ is the piece of wire ran between the grounded conductor and a lug, bar or other termination device, correct?
I don't think anything in the code says what color the MBJ needs to be. Regarding your second sentence, it depends what that lug, bar, or termination device
is.
Here is my point regarding the OP's question. I can imagine four scenarios in a service panel that has two separate bars.
1) One of the bars is isolated from the enclosure, and the other is not. A wire, or factory installed bar, runs between them; that is the MBJ. You are not supposed to land neutrals on the non-isolated bar, which we would call the equipment ground bar.
2) Same scenario as above, except there is also a bond (green screw, strap, whatever) on the insulated bar. This might happen if the panel was only manufactured with one bar, but someone added the non-isolated bar to gain more terminals. In this case it's debatable which object is the MBJ, but you definitely can't put a neutral on the non-isolated 'equipment ground bar' because you'll have objectionable current on the enclosure.
3) Green screw or strap or wire - i.e. MBJ - bonding the neutral bar to enclosure. Separately, a non-isolated 'equipment ground bar' is bonded only directly to the enclosure. Enclosure serves as the EGC connecting the 'ground bar' to the MBJ.
4) Two isolated bars with a factory installed insulated bar connecting them. An MBJ connects one of them to the enclosure. In this case both bars are 'neutral bars'. It is not appropriate or necessary for one to be labeled 'neutral' and the other to be labeled 'equipment ground'. (I don't think it is a violation to land an EGC on them, however.)
In
none of these four scenarios is it allowed to land a neutral on something which is appropriately called an 'equipment ground bar.'