I do not know the reason for using symbols but I guess there must be. Personally, what you are asking can just be accomplished by using timelines with in the Hype document. But, as you have mentioned you wanted to use a symbol then the following may be useful.
Whenever you create a symbol Hype automatically creates a “symbol action”
that plays the main timeline of the symbol. If you look in the main timeline of the Hype document you will see it. You just have to delete it to stop the symbol from playing automatically.
That being said you could just create a timeline in the main doc and attach it to the mouse over and mouse down actions in the inspector. No need for a symbol.