Need to amplify a square wave, 20 amps or so
a square wave (normally unipolar) sine, triangle

So this is a set of three different waveforms you need to amplify.
Problem is that square and triangular waves have theoretically infinite bandwidth. So, in reality, they are limited to some finite slew rate – you'll need to figure out what that is, but I'm pretty certain it's above audio frequency levels.

Is there any reason this wouldn't work?

Plenty. An amplifier is always bandwidth-limited, and especially a class-D amplifier only works by smoothing out a square wave until it doesn't look square anymore.
So, you want the actual power transistor part of that amplifier, but nothing else of that amplifier.
In other words: getting a set of MOSFETs to do the switching is exactly the right (and only) thing you should do if you need to amplify a square wave.
For a sine wave, things look drastically different. You need an amplifier with a linear in/output relationship – and at 5V/20A, ie. 100W, that's a pretty beefy amplifier. Looking at that ebay offer, there's positively no way imaginable the amplifier depicted comes close to 600W output power. Maybe 150W, or so. And that doesn't mean anything's sufficiently linear at 100W.

One problem might be(?) that I want to use a square wave, will the dc filtering remove this?

Possibly, but what's worse is that the output filter of that amplifier will never allow you to get sufficiently steep signal edges.

Does uni vs bipolar square make a difference?

Depends on how the wiring of your ignition actually looks like. The spark as is doesn't care, but in reality, the vehicle ground usually closes the circuit, so you typically can't choose an arbitrary virtual ground, and hence would need your amplifier or its power supply to generate a negative, 20A-average-capable supply – that's a hard job. So, I guess it's most likely you'd want to end up unipolar.
Conclusion
It's always &quot;easiest&quot; to switch something on and off – it's where transistors waste the least power. If you want to drive ignition coils, also, you typically want the highest \$\frac{d\,i(t)}{dt}\$ possible, so, you'd also want hard switching in that case, and not a sine wave or anything else. You're not trying to generate a high-voltage supply, you're trying to generate short sparks, so getting a nice smooth power transition from primary to secondary side in these transformers isn't at all what you want – I don't know, but I guess the transformers thermally aren't designed to continously transport power, but really just short impulses.