Without being able to see the cables as they enter the cabinet; or the ability to touch or trace them, here is what I assume is going on.
Definitions:
Grounded (neutral) from the service
A typical single split phase service is made up of 3 wires. Two ungrounded (hot) conductors, and one grounded (neutral) conductor. The ungrounded (hot) conductors will connect to the main service panel through a disconnect (usually a large breaker), while the grounded (neutral) connects to the neutral lug. The neutral lug will be bonded (electrically connected) to the neutral bus bar, and all grounded (neutral) branch circuit conductors will terminate at the neutral bus.
Grounding Electrode Conductor
This conductor is used to connect the grounding electrode (ground rod, etc.), to the grounding bus in the panel. All equipment grounding conductors will be connected to this bus.
Bonding Jumper
The bonding jumper is used to bond (electrically connect), the un-energized metal parts of the panel to the grounding system.
Assumption:
Since it appears that (what I assume is) the grounding electrode conductor terminates at the neutral bus, I'm also assuming that this is the main service disconnect. This leads me to believe that the neutral and grounding buses are bonded (electrically connected). In which case, technically, grounded (neutral) branch circuit conductors can terminate at the grounding bus.
So you have two options:
Terminate the grounded (neutral) from the new circuit to the grounding bus.
Move the green wire that is terminated on the neutral bus, to the grounding bus. Then terminate the grounded (neutral) from the new circuit, to the freed up slot on the neutral bus.
Additional Information and Code Compliance:
Number of Conductors
Since this is a new circuit, it has to be installed to current code standards.
National Electrical Code 2011
ARTICLE 250 — GROUNDING AND BONDING
VI. Equipment Grounding and Equipment Grounding Conductors
250.140 Frames of Ranges and Clothes Dryers. Frames of electric ranges, wall-mounted ovens, counter-mounted cooking units, clothes dryers, and outlet or junction boxes that are part of the circuit for these appliances shall be connected to the equipment grounding conductor in the manner specified by 250.134 or 250.138.
Which in this case means installing a NEMA 14 receptacle for the dryer, and a proper grounding conductor.
You'll have to follow the dryer manufacturers installation instructions for upgrading to a 4 wire cord. For more information see this answer, and this answer.
Since you've said that you're already using 4 wire cable, you'll simply have to terminate the grounding conductor in the cable to the grounding bus in the service panel. Then connect the other end of the grounding conductor to the grounding terminal in the dryer receptacle.
Size of Conductors
You'll also want to be sure that you're using the proper size breaker and conductors. In the case of a dryer, you'll typically use a 30 ampere breaker and 10 AWG conductors (depending on the length of the run). However, you'll want to check the dryer manufacturers installation instructions to verify this.
From your description it sounds like you have the intelligence to understand this stuff, but are simply lacking certain nuggets of information. You'll want to read up a lot on "switch loops" and particularly "3-way switches". It will all make sense pretty quickly, then.
Oh, and one more thing that's a bit harder to uncover: In America, wire colors do not have firm meanings. Green always means ground. White or gray always mean neutral, unless marked or in switch loops and messengers - the goal being to build the necessary circuits with readily available multi-wire cable even though it's the wrong color. "Hot" lines are everything else - including marked whites and (implied) switch-loop whites, and there is no official standard of how to designate certain types of hot. (however in commercial facilities, consistency is required.)
Best Answer
DANGER!
This is totally wrong. You have 2 pairs of 1/2-size breakers - a.k.a., "double stuff" or "tandem". If you measure voltage between black and red (i.e., between a pair of hots) you will find 0V instead of 240V. If this was done by an actual licensed electrician, he should have his license revoked. If it was done by a handyman-acting-as-electrician, this is why most places don't allow that.
Note that this would be perfectly fine with two separate cables - i.e., each breaker getting its own black & white wires. But sharing (MWBC) only works if the pair of breakers are (a) on opposite poles/legs so that you get 240V between the hots and (b) the breakers are physically paired (handle tie or double-breaker) like the 40A pair across from these breakers.
And now for the really crazy part. The usual reason to go for 1/2 size breakers is because you don't have enough space to put full size breakers. But you have more space right
belowabove (picture is upside down, or panel is upside down) the new breakers! You could easily have 2 pairs (i.e., 2 double breakers) of 15A full-size breakers and everything would be just fine.There is actually one other alternative. A quadplex breaker:
The beauty of a quadplex is that the inner and outer pairs each function like a traditional double breaker. Red and black to the outer breakers + white neutral. Red and black to the inner breakers + white neutral. Done.
Difference in cost between 1 quadplex and 2 tandem breakers at Home Depot? Less than $11. Probably even less of a difference at an electrical supply house.