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I would suggest to change three thinks which haven't been mentioned yet.

1) Your regex works but is not what I call "easy on the eyes". What about the following:

^[0-9]*$

When I read something like this I immediately think: "Some regex with number filters."

2) Also same topic. Do you actually want to allow empty inputs? I would suggest not. Replace * with +:

^[0-9]+$
^\\d+$

3) I like static imports. It is not everyones favourite but in my opnion it makes the code more readable. That is why this would be my final solution:

import java.util.Optional;
import static java.util.stream.Collectors.joining;

public class ShoutySnake {

    private enum WORDS{ZERO,ONE,TWO,THREE,FOUR,FIVE,SIX,SEVEN,EIGHT,NINE}

    private static final String NUMBERS_ONLY = "^[0-9]+$";

    private static String getNumericalCharAsSnakeShoutCaseWord(final int codePoint){
        return WORDS.values()[Character.getNumericValue(codePoint)].name();
    }

    public static String convertNumbersToShoutySnakeCaseWords(final String input) {
        return Optional.ofNullable(input)
                .filter(s -> s.matches(NUMBERS_ONLY))
                .orElseThrow(() -> new IllegalArgumentException("input cannot be null, empty or non-numerical."))
                .chars()
                .mapToObj(ShoutySnake::getNumericalCharAsSnakeShoutCaseWord)
                .collect(joining("_"));
    }
}