Paid plugin also available
If you're frustrated with tutorials and want a quick solution that definitely works, check out Squarespace Click Traking Plugin.
It's a paid solution, but a few dollars also makes it 100% sure to work. Money back if it doesn't and 1-on-1 support provided if you have any problems.
The Squarespace Button Block is basically a link as any other, so link tracking is what you need. And it's what our plugin does. The plugin tracks outbound links, internal links, file downloads, button links, social media links etc.