Is it possible to print a receipt to a square-supported receipt printer using the register API instead of the square app itself? And if not, how would I interface with some of these devices?
This is not possible directly via the Register API at this time, although - as you mention - you can use the Square app itself to connect to a receipt printer and print receipts. To interface directly from your app to a printer, you can look at provider such as Star Micronics which has Android and iOS SDKs (http://www.starmicronics.com/support/sdkdocumentation.aspx)