I think some context would be useful! So the best answer I can give is to say it depends on the rest of the sentence. My first reaction was to say boldly, they are both wrong. However...
It's easy to make a case for "either clerk", or even "either clerks" (without an apostrophe): If it is a choice between two clerks, then you use the singluar, for example: "I didn't trust the job to either clerk." If you go on to say "or...", the plural can sometimes be appropriate: "all the staff in the office were either clerks or interns".
The trouble is, there's no noun in the given phrases; as modifiers, "clerk's" and "clerks'" need a noun to pertain to.
Having said they are both wrong, I can tell you that they are (potentially) both right. It's all about context.
An example for the first one would be: "Nobody was sitting at either clerk's desk" (two clerks, absent from two desks). The second is a bit trickier, but not impossible: "I concluded that either clerks' imaginations were lamentable in general, or it was this dreary place that stifled them" (two conclusions; one about clerks, the other about the place).