ForEach(0..<3) {
    Button {
        //code
    } label: {
        Image(countries[$0])
    }
}

This doesn't work because the initializer you are using for Button is this one:

init(
    action: @escaping () -> Void,
    label: () -> Label
)

Note that the action and label parameters are closures, which means that $0 you are trying to use in label actually refers to the first parameter passed into that closure, not the parameter of the ForEach. Except, hang on, the action and label closures do not take any parameters. And so, you get an error.

You can see that $0 doesn't belong to the ForEach if you add a line like so:

ForEach(0..<3) {
    let _ = print($0)
    Button {
        //code
    } label: {
        Image(countries[$0])
    }
}

When you do that, this error will disappear: Contextual type for closure argument list expects 1 argument, which cannot be implicitly ignored because you are no longer ignoring the parameter to the ForEach.

It works here:

ForEach(0..<3) {
    Text("\($0)")
}

because Text is taking in either a LocalizedStringKey or a String rather than a closure, so $0 in this case refers to the parameter of the ForEach as you would expect.

This code:

ForEach(0..<3) {_ in
    Button {
        //code
    } label: {
      Image(countries[_])
        .renderingMode(.original)
    }
}

doesn't work for a couple of reasons.

First is because of the same reason the first example above didn't work; you are trying to reference a parameter passed to the label closure but that closure doesn't take any parameters.

Second, and most importantly, you can't access an array this:

countries[_]

Ultimately, you end up having to do this:

ForEach(0..<3) { number in
    Button {
        //code
    } label: {
      Image(countries[number])
        .renderingMode(.original)
    }
}

Hope that helps explain what's going on.