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.