Hello
Would a Canon Speedlite 580ex ii flash work on a Nikon D3400?
This can certainly "work", as in "function", but because the Canon and Nikon hotshot contacts are different (apart from the center pin) and because the TTL "language" is different, you'll only be able to use the Canon flash in manual mode on your Nikon.
It can work as an external with the SB-600, but not via Nikon CLS...you'll need to use a PC cable or optical trigger or other means of firing it (PocketWizard or similar).
This can certainly "work", as in "function", but because the Canon and Nikon hotshoe contacts are different (apart from the center pin) and because the TTL "language" is different, you'll only be able to use the Canon flash in manual mode on your Nikon.
There's certainly nothing wrong with that if you know how to use manual mode, but there are cheaper ways to get a manual-only flash with comparable power.