java.lang.String is declared as final, however are there any mechanisms available legitimate or otherwise to extend it and replace the equals(String other) method?
No, absolutely not. If you want some "other" kind of string, create another type which might contain a string:
public final class OtherString {
private final String underlyingString;
public OtherString(String underlyingString) {
this.underlyingString = underlyingString;
}
// Override equals however you want here
}