Which is correct or both are correct?
Art students can either take a non-science elective or a math elective.
Art students can either take non-science electives or math electives.
If they are required to take an elective, and if that required elective can be either math or non-science then you could word it as:
Students shall take either a math or non-science elective.
If you’re just trying to say that they are allowed to take an elective (in addition to other courses), either of your original forms are acceptable. However you should use MAY and CAN.
Is there any difference in meaning between
girls may get a book
and
girls may get books
:?:
The second one means that girls are allowed to obtain/receive books (in general) without specifying how many or whether a specific book is involved.
The first one implies that they are allowed to get a single book (say, out of a stack of books on the shelf).