Broke a leaf spring. Shop says I should instal bigger ones?

I would upgrade all the components just as others have said. There have been many reports of broken springs on various makes of trailers. Not enough carrying capacity margin with original factory springs.

However, I wouldn't go overboard. Go only to the next higher rating spring sets than you originally have. Any thing higher than that may be too stiff resulting in a hash ride beating up your trailer as Ducttape pointed out.