There are a few things to consider here. First the phrasing of the question. There should have been some punctuation or formatting to clarify. That is, if the question was multiple choice, it could be:
The eigenvalues of A are:
(a) either 0 or 1
(b) both (0 and 1)
Alternatively, The question could mean:
The eigenvalues of A are either:
(c) both (0 and 1)
The answer you said the exam board gave implies the first question was asked.
The second thing to consider is the logical implications of "or" vs "and". The use of the word "either" implies an "exclusive or", that is the answer can be 0 or it can be 1 but it cannot be both. The word "and" means that it must be both 0 AND 1 (it cannot be one and not the other).
It sounds like you and the exam board read the question differently. You may have interpreted the question as the second example above and therefore selected the most sensible option (the eigenvectors could be both, but not at the same time). If the exam question is worded exactly as in your post, it does seem a little unclear if not ambiguous. I suspect (hope!), however, it may have had some punctuation (like semicolons) which would clarify it.