Yes you can....then theres the but. It depends on where you are and the demographics of your particular unit and so on. Best person to ask is the BM 1st class/ chief at your unit if you are interested.
Yes you can become a COXN at smallboat unit MSST , but it would be up to the OINC to let you certify. It is also done seperate from your normal duties at your unit. But heck engineers make better boat handlers, so go for it.
I'm an MK1 and certified Coxn on both the 41 and 25. First things first, ensure that you can perform your current duties without letting driving boats get in the way, (putting your primary duties on the back burner), second when you make Coxn, U/W, don't play engineer, let them do there job for the experiance. In the end its a great feeling.