When Does a Shift to the Right Happen in the Demand and Supply Curves Simultaneously? If, for example, Health Canada issues a public statement that states eating pizza is good for your overall physical health, would this result in both a shift in the demand curve for pizza to the right, and a shift in the supply curve of pizza to the right?

If not, could you give more information on when this would actually happen?
To answer your first question: No, it shifts out only the demand curve but not the supply curve. However, the quantity supplied increases.

To understand why the advertisement does only shift out the demand curve but not the supply, you have to understand two important concepts: 

(1) Shift of the demand curve = Increase of demand;
(2) Movement along the demand curve = Increase of quantity demanded.

Similarly with supply:

(1) Shift of supply curve = Increase of supply;
(2) Movement along the supply curve = Increase of quantity supplied.

Now what happens, if you advertise for pizza? The demand as well as the quantity supplied increase. To see this, consider the following. More people demand pizza so the demand curve shift out and the demand increases. At the same time, the supply curve does not move. BUT, the outwards shift of the demand curve intersects the supply curve further to the right, and hence, the quantity supplied increases.

Shifts of the demand curve can be caused, for example, when people's income increases or decreases. Demand increases usually increase the quantity supplied.

Shifts of the supply curve can be caused, for example, through the the use of better technology that lowers producers marginal production costs. Supply increase usually increases the quantity demanded. 

Increases or decreases of quantity demanded or quantity supplied can be caused through price changes.

A simultaneous increase of demand and supply can be cause if consumers get a higher income and firms get a better technology at the same time.