From what I've seen (which isn't all that much), horn buttons typically ground the circuit to complete it, where power buttons typically complete the positive connection. Seems like it wouldn't matter but all of the horn buttons I've worked on only have one wire running to them, the ground is completed through the body of the switch. Point being I wouldn't run a positive lead to the horn button until I was sure it was isolated from ground. And I've never seen a horn button that's isolated from ground. They may be fairly common, I don't know, but it's certainly not a given it will work on any bike. Some kill switches work the same way FWIW.
As for wires being large enough, use a relay. they're cheap.
"Funny" story. When I was figuring out how to lay out the wiring on my 750GT, I spent a bit of time trying to figure out where to put the starter relay. I considered a few different places, then tried to remember where the stock one had been. I couldn't remember. Then it occurred to me that a 750 GT is kick start only. D'oh!
EDIT: You could still use the horn button to trigger the starter relay, but you might(?) have to do it by breaking/completing the ground, not the positive side. That's how the horn on my 750 works after all, no reason it couldn't be a starter motor it was firing instead I suppose. The wiring would be pretty simple but I'd strongly urge you to get a wiring schematic and figure it out/ask specific questions rather than have someone tell you how and not understand what you're doing. It'll make the trouble shooting that much easier if/when you start letting the smoke out of the system.
EDIT 2: I'm assuming by ignition button you really mean starter button? As in it turns the starter motor, not turns on the ignition circuit?