The first and the last sentence are correct. The middle sentence is not correct.
The reason is this sentence, which is the original one:
- There should be an X chromosome and a Y chromosome.
Notice that this is not
- *There should be an X chromosomes and a Y chromosomes.
Neither chromosome should be plural.
That's what is meant.
Now the rule of Conjunction Reduction deletes the first chromosome, leaving sentence 1
- There should be an X [...] and a Y chromosome.
Since chromosome wasn't plural before Conjunction Reduction, it isn't plural afterwards.
So the second sentence above is ungrammatical.
Conjunction Reduction only deletes; it doesn't do arithmetic.
However, the speaker can do the arithmetic.
There is, after all, one X chromosome and one Y chromosome involved,
and that makes two chromosomes, should one need to speak of them.
(There is, of course, no article, since a/an is only singular.)
But X and Y is a perfectly reasonable conjoined NP that can modify
a plural chromosomes, which leads to the third sentence:
- There should be X and Y chromosomes.