Just another question to make sure I don't make any mistake. I am going to install 3 Florentine lights but I need to extend the lead wire about 10 feet each using hot shrink, and then connect the leads to the run or Pro Junction Hub. Is 14/2 acceptable?
With what you described, it doesn't sound like you are making any extremely long runs off of the hub (which can lead to voltage loss with multiple fixtures hooked in) so, you should have no problems with doing those extensions. The gauge on the lead wires is 18AWG just as an FYI for the type/size of connector you will use!
Put the shrink tubing over the wire first, then use a butt connector, pack a little electrostatic grease in the ends of the connector, crimp it, tug it to make sure you have a good connection, rub a little more grease around the ends of the connector, and in the tubing, heat the tubing from the middle first. If the grease pushes out both ends as you heat from the middle, the connection will be relatively corrosion proof for years.