UK and Ireland - 2017 - 2021 (some features are restricted for 2017)
Canada - Sage 50 Pro*, Premium & Quantum, version 2019.1 and after
*For Sage 50 Canada Pro Edition, please see here for specific restrictions.
For publishing invoices, through to QuickBooks there is no restriction on the version/year of QuickBooks Desktop as far as we are aware. We have suspicions that versions older than 2013 will not work.
For statement data, AutoEntry provides a Web Connect File type specifically for QuickBooks Online and Desktop. The Sync app is not required for QuickBooks Desktop if you only statement data through AutoEntry. All versions of QuickBooks Desktop 2016 - 2018 can accept these files. Intuit themselves restrict the use of web connect files to older versions of QuickBooks Desktop.