Navigo passes are valid on all trains that don't have compulsory reservation, regardless of whether the train is a Transilien (Île-de-France regional train), TER (regional train from another region), or Intercités (cross-region train). So that part isn't a problem. The problem is that the Navigo pass is only valid within Île-de-France. Even on a Transilien, the Navigo pass isn't valid for journeys outside the administrative boundaries of Île-de-France. Rail maps indicate the boundaries, for example on the official Transilien map the outside part with a dark background is outside Île-de-France (“Tarification hors Île-de-France” means “fare outside Île-de-France”).
In France, in most cases, you need a single fare for your complete journey. You can't combine different tickets or passes at a border between zones. There are a few exceptions, but they don't apply here: a Navigo pass can't be combined with a point-to-point ticket¹.
If you're going from A to C with a change of trains at B, then of course you can use a ticket or pass from A to B and a different ticket or pass from B to C. But you need to validate the second ticket at B. If you take a train from A to C and the train calls at B, you can still do this, but in principle you need to get off the train. In practice, I doubt that ticket collectors would enforce the rule so zealously, you can stay on the train if you don't need to validate anything at B.
For your trip out from Paris to Vernon, almost all trains call at Mantes, and reservations are not compulsory. So you can use your Navigo pass from Paris to Mantes, and get a ticket from Mantes to Vernon. If you get a paper ticket, you'll need to stamp it in Mantes: if you stamp it in Paris and your ticket is checked at Mantes, there's a serious risk that you'll get a fine because your ticket isn't valid. On the other hand, with a smartphone ticket, there's nothing to validate. So if you get a smartphone ticket for the Mantes–Vernon part, you're good.
For the trip back from Vernon to Paris, in principle, you need to validate your Navigo pass in Mantes. The pass stores the location and time of last validation. However, the fine is 5€ if you pay on the spot, and that's less than the fare difference between Vernon–Paris and Vernon–Mantes. Furthermore, in my experience, ticket checkers in SNCF mainline trains don't check the location of last validation (I'm not even sure if their machine shows this information).
So get the Mantes↔Vernon tickets on a smartphone and you're good. (Assuming your train does call at Mantes.)
Part of your question reflects obsolete or incorrect information about Navigo passes. Unfortunately many websites, even some official ones, aren't up-to-date.
Since 2015, all Navigo passes cover the whole Île-de-France region, except for two-zone passes that don't include Paris. So if you're based in Paris, zone-free days aren't relevant.
There used to be extension tickets for Navigo pass holders, priced at the fare from the last station within the zone of validity to the destination station (when starting from within the zone of validity; or symmetrically in the other direction). These tickets no longer exist since the 2015 rate changes that caused almost all passes to be valid for the whole region in the first place. And anyway extension tickets were only ever valid within Île-de-France, never for journeys extending past the regional border.
¹ It's possible to combine a Navigo pass with some main line passes that run up to the first station within the Navigo area, but those are monthly or yearly passes priced for commuters who do a return journey every weekday.
