About your primary question: I'd definitely go with Answer... whenever I have the possibility to do so. As you can pass on an array to a varargs parameter, but not varargs to an array parameter, varargs is the better option IMO.

Other comments

Naming

private boolean isCorrect;

It is convention to not prefix boolean fields with is. The name of the method should be is + variable name with upper case starting letter, so:

public boolean isCorrect() {
    return correct;
}

final keyword

It is good practice to mark all possible fields with final, this applies in your case to:

private final String answer;
private final boolean correct; // note the name change

and in the Question class:

private final String question;
private final Answer[] answers;

Mutability

This code might seem harmless to you:

public Question(String question, Answer... answers) {
    this.question = question;
    this.answers = answers;
}

public Answer[] getAnswers() {
    return answers;
}

But the fact is, this is slightly dangerous. Although your Answer class is immutable, this is not. Imagine the following code:

Question questionA = new Question("Do chickens fly?",
     new Answer("Yes", true),
     new Answer("No", false));
questionA.getAnswers()[0] = null;
questionA.getAnswers()[1] = new Answer("Always right", true);

Whoops, now we've cleared one answer and changed the other!

Solution: Make array copies using Arrays.copyOf:

public Question(String question, Answer... answers) {
    this.question = question;
    this.answers = Arrays.copyOf(answers, answers.length);
}

public Answer[] getAnswers() {
    return Arrays.copyOf(answers, answers.length);
}

Possible extension: Make it a score

Your classes supports multiple answers to be correct, that is nicely done. But what if one answer is still "better" than the other? My suggestion is that you use a int or a float for score instead of a boolean correct.

Java 8

If you have Java 8 available, your isCorrect(String answer) method can be improved:

public boolean isCorrect(String answer) {
    return Arrays.stream(answers).anyMatch(a -> a.getAnswer().equals(answer));
}

The same can of course be applied to your isCorrectIgnoreCase method:

public boolean isCorrect(String answer) {
    return Arrays.stream(answers).anyMatch(a -> a.getAnswer().equalsIgnoreCase(answer));
}