The way I would do this is to use the squareup.com webpage to upload a catalog of Bid numbers. If you were doing this for a auction once a week I would create a google sheet like this: https://goo.gl/9Z5WoQ (ask for permission to edit the file if you want to interact with the sample file).
The item name column contains the date of the auction and the bid number as B0001, B0002, etc. Likely you print up only so many bid number tickets so I would create the item name column to have as many rows as you need. Using a spreadsheet trick, highlight row2 (columnA-L) and take the corner and drag it down to automatically create as many rows as you going to print bid tickets. Do not put values or change the Token column, and by leaving the Price column blank this will allow you to place the final Price in Register at the time of checkout. Put the appropiate sales tax in column L.
In playing with this I though it might be useful to have the Catagory column be the date of your Auction, that way you can create a tile on your register by date. The date has a period at the end of that string to prevent the date from incrementing when you create additional rows, see above. But I still think its important to have the date of the auction imbedded in your item name too. The reasons for the date to be associated with your auction is that item names that are identical create an items with variations, and this is likely not what you want in squareRegister.
Once you you create this sheet, you will need to download it as a csv, in Google Sheets, use File -> download_as -> Comma seperated values. Then on the Squareup.com website, go to Items and Import this file.
Once imported to square, on Register you can now check a bidder out either by doing a Search by bid number "B0012" or finding the bid number in the pull down by that catagory.
