Does “either a or b ” preclude “both a and b”?

No., Either A and B, do not preclude both A and B.
Either means any of the 2. Thus when you say, either a and b.
meaning the answer may be A or B.
But when you say Both A and B.
Thus this simply means that the answer is both A and B.

=> Either is not the same as both.
Therefore the answer is NO.

For example:
When are you coming:
Answer = either today or tomorrow, meaning your coming maybe today or may be tomorrow. You're not sure.