Where is the button located to cancel a bid? It's somewhere on the listings page but I can't see it anywhere.
A bid may be retracted (canceled) by the buyer in some cases if the retraction meets the requirements of our Bid Retraction policy. http://pages.ebay.com/help/buy/questions/retract-bid.html
Correct. The only circumstances in which you can request that your bid is cancelled are; You make a typographical error and enter the wrong bid amount. For instance, you bid $99.50 instead of $9.95. If this occurs, enter the correct bid amount immediately after you retract your bid. If you do not place another bid, the retraction will be in violation of eBay's policy and could result in your suspension. Please review the retraction guidelines if you need to retract your bid. The description of an item you have bid on changed significantly after you placed your bid. You cannot contact the seller. This means that you tried to call the seller, but his or her phone number doesn't work, or that you have tried emailing a message to the seller and it comes back undeliverable. All of the above have been taken from the page I linked in my original post.
Ask them why they want to cancel their bid. If they give a reason other than the 3 I listed above tell them it isn't possible for you to cancel their bid and point them in the direction of the link I showed you. It explains everything.
If you cancel their bid (which is doable) then it will not show on their record that they have a bid retraction. Refuse to cancel the bid and tell them a bid retraction by the bidder is the correct way to handle it.