They don't have a link to the actual form the counties want to distribute, but when Republicans get tweaked over this, it's usually because they are using the Federal mail-in form. They ask on that form that people simply check a box to indicate they are US citizens and are thus eligible to vote. Since it's a mail-in form, they don't ask to include any other proof of citizenship (although first-time registrants do need to submit proof of residence). But, they do note on the form that submitting a form with a false declaration is a crime, and if a non-citizen does it they would be subject to deportation.
If Paxton is so sure non-citizens are registering to vote in droves, all he has to do is investigate these forms as they come in, as any non-citizen who fills it out can immediately have the book thrown at them. The fact that he doesn't means that even he knows non-citizens are not doing this. It makes no sense to risk being kicked out of the country just to vote in a system where you can't fully participate. The risk is just too high.