The code is quite complicated. You could just do this:

public static boolean equalsIgnoreNewlineStyle(String s1, String s2) {
    return s1 != null && s2 != null && normalizeLineEnds(s1).equals(normalizeLineEnds(s2);
}

private static String normalizeLineEnds(String s) {
    return s.replace("\r\n", "\n").replace('\r', '\n');
}

The word twirks sounded negative to me, therefore I replaced it with style.

Since the two strings have equal rights and are treated the same, none of them should be called the "other one".

The code looks quite complicated at first glance, but when reading it, it is straightforward. If you want to make the code shorter, you can just do this:

public static boolean equalsIgnoreNewlineStyle(String s1, String s2) {
    return s1 != null && s2 != null && normalizeLineEnds(s1).equals(normalizeLineEnds(s2));
}

private static String normalizeLineEnds(String s) {
    return s.replace("\r\n", "\n").replace('\r', '\n');
}

Concerning running time and GC stress, your code is probably better. Use a benchmark to see how much better it is.

The word twirks sounded negative to me, therefore I replaced it with style.

Since the two strings have equal rights and are treated the same, none of them should be called the "other one".

You should not make the constants public, since there is no need to do that.