I was given a 10-2 wire as it is only one unit directly to the panel and a 30/30 amp breaker. However, the unit only has a black and white wire and the cooper ground. After connecting the white, the black and the ground, I still have an empty 30amp and I am wondering if they gave me the wrong breaker or if I should have 10-3 wire.

For a 240 volt circuit using black and white, both wires need to connect to the breaker, black to one terminal and white to the other.

This will deliver 240 volts to your heater.

I suspect you connected the white to the neutral bar and black to one terminal on the breaker, thus leaving an unused terminal on the breaker.

When 2 wire cable with a white wire is used for an energized conductor, and not for a neutral, the white wire needs to be permenently identified at both ends that is not a neutral and that it is an ungrounded (live) conductor. A permenent black marker pen works.