Is your car an early '65 (what we call a '64.5) or is it a '65? The way you describe the horn wiring it sounds like a '65.
There are 2 spring loaded contacts that are part of the turn signal switch below the steering wheel. One of them is the "hot" wire and the other one goes to the horns. The entire purpose of the horn button is to connect these 2 contacts together sending power to the horns and causing them to blow. The horns are grounded where they bolt to the body so there is no ground that is part of the horn button.

 

I do have the ‘65 style with the two spring loaded contacts that touch the backside of the wheel. I had viewed the purpose of the horn button as a way to complete the circuit between the hot terminal and the one that feeds the horns. It’s essentially acting as a switch.
What I should have added in my original post is that there is a metal clip that is spring loaded that holds the horn button into the wheel. It’s the when the metal clip on the button contacts the hub of the wheel that I have problems. This is where my spark is coming from as I have 12V going through the clip.

 

Another member was having that problem recently. The solution was something about some screws in the steering wheel being too long or having heads that were too large. More chinese crap.

 

There is no ground at the horn button on ALTERNATOR-equipped Mustangs. The horn button is simply a SPST (single pole-single throw) momentary switch that completes a circuit, directing the power received back down the steering column and out to the horns, which each are grounded through the chassis.

 

I took the horn button apart, please see the attached pics.

From what I can gather, the spring that makes the horn button pop back up is wedged under the plate that connects the 12V wire to the lead going to the horn. The tab that holds the button in place is part of this spring and is electrifying when I push the button. It needs to be separated and shouldn’t be touching the hot wire. It actually has the ground soldered to it but it’s getting powered up instead.

 

Update: after disassembly, I put the horn button back together with the spring repositioned so that it couldn’t touch the internal horn contacts. Once hooking it back up, everything works perfectly now. I have the two leads from the steering wheel hooked to the two posts on the back of the horn button. I clipped off the extra ground wire as it’s not used for this setup. Thanks to everyone for their help, I had assumed that the horn button was the only thing that was correct in this equation.

 

I had this same problem a few days ago.
I had used a battery charger as a power supply to test every wiring harness when it was installed.
Every light, turn signal, fan, etc., worked when tested.
After I installed the steering wheel, I got a massive current draw when using the power supply.
Wondering what was wrong, I remembered that the steering wheel was the last thing I installed, and since I didn't have the horns, they weren't tested.
I pulled the horn button off, and the diameter of the spring is large enough that it contacts the spade terminals on the button, and the wheel hub, causing a dead short.
I had some small diameter heat shrink tubing, and worked it onto the base of the spring, thereby covering the spring and preventing it from shorting out. To play it safe, I covered the base and top of the spring with heat shrink.