The changes cannot be saved to the system definition, because the entered connection properties contain invalid values.

​RevertToSelf, so named because of the method call used to implement it, is an authentication method in which the identity of the process running BCS is used to access the back-end system (usually a database such as SQL Server). It is also known as "BDC Identity" in the dialogs of SharePoint Designer. This allows you to use the trusted subsystem model of authentication against a back-end system.

Whether you should do this at all on a production server is perhaps another matter for security experts to debate. Anyway, for whatever reason, that is what you are trying to do. Seems simple enough - what could possibly go wrong?

I turns out this is disabled by default (are they trying to tell you something?). To enable it, you need to do a bit of work with PowerShell on the server, thus:

$bdc = Get-SPServiceApplication | where {$_ -match "Business Data Connectivity"}
$bdc.RevertToSelfAllowed = $true
$bdc.Update