This box you want to use should have had a normal switch to control the lights. Circuit breakers are normally not designed to be constantly toggled on and off. Once you get this timer going, all should be good.
The problem is this box currently only has hot wires in it, there is no neutral that the timer needs. The black should come directly from the breaker. The white should run to the lights themselves, hopefully tied into a black wire in the breaker panel going to the lights. The wire returning from the lights is the neutral and should be white coming into the breaker box. There's a slight chance this was done backwards.
In order to get a neutral to the timer, you need to remove the metal cover to the breaker panel. WARNING! There are live exposed metal parts in there that will electrocute you if you contact them. Turning off the main breakers will kill power to most of these parts, but the large wires connecting to the main breakers will still be hot. To be totally safe, your electric meter needs to be pulled out of it's base. (TBH, many knowledgeable people will do this sort of work with the large wires hot) Not only has the power company sealed the ring that prevents meter removal, but some types of bases require another panel removal, exposing more live parts. Thus you should have the power company pull the meter for your work (or otherwise kill all power into the breaker panel) and replace it when done, to be totally safe. And even with the meter out, some of the jaws in the meter base will remain hot. This is dangerous stuff if you don't know what you're doing. The power company may refuse to help you unless you do sound like you know what you're doing.
I'm assuming this is your main breaker panel. If it is a sub-panel, just turn off the breakers feeding it from the main panel. No need for power company involvement. If you have the slightest doubt about your ability to do this work, do not hesitate to engage a qualified electrician. Your health is worth far more than what any electrician would charge.
Once the breaker panel is made safe and you remove the cover, confirm the white wire in the switch box does indeed connect to the wire going to the lights. If so, mark this wire at both ends with black tape to indicate it is a switched wire and not a neutral.
Now locate the neutral bar. It has many binding screws and many white wires connected to it. There may be more than one. Obtain a length of solid copper white wire of the same gauge as the other lighting wire. If the breaker is 15 amp, it is likely 14 gauge. A 20 amp breaker would usually have 12 gauge. The length should be enough to run from the (closest) neutral bar into the switch box with about an extra 6 inches.
Connect one end to any empty binding screw on the neutral bar. Push it into the switch box and connect the other end to the white wire of the timer. Fold the wire so it is routed neatly inside the panel. Connect the black wire from the breaker to the black wire of the timer. Connect the white wire you marked with black tape to the timer's blue wire.
Fasten the timer to the switch box and install the cover plate. Replace the cover to the breaker panel. Restore power. Use the timer's manual switch to verify it correctly controls the lights. Follow the instructions to program your timer. That's it!