l wasn't able to find exact numbers on the increase of Shiny rate the DexNav increases with each chain, so these calculations will be based on [this test l found on Reddit though][1], having a higher chain or higher search level doesn't increase the rate of encountering shiny Pokemon.
The chance of encountering a Shiny Pokemon with DexNav according to the document l found (with an average of 308.3333333 search level and 95.23188406 chain level) is within the range of 1/308 and 1/512. (this was the only data l could find about this rate)
The probability of encountering a Pokemon with all max IVs naturally is 1/32^(6) or 1/1073741824 however the DexNav is able to leave you to encounter Pokemon with the Potential rating system up to 3 stars (or 3 perfect IVs). l wasn't able to find the rate of getting 0, 1, 2, or 3 stars, but l'm going to base the calculation on the positive side of the spectrum.
Basing this variable on the assumption that this Pokemon will always have 3 stars, will have a rate of getting a Pokemon with all max IVs is 1/32^(6-3) or 1/32768.
The rate of getting a Pokemon with a Hidden Ability according to an observation by Bulbapedia is 23%(23/100) with a search level of 100, and 5%(5/100) with a search level of 10.
Taking this mess of junk and numbers:
The rate of getting a Shiny Pokemon with all Max IVs and Hidden Ability
(with 0 stars of Potential, and Lv 10 search level and using the minimum probability of getting Shiny based on the test 1/512)
1/512 1/1073741824 5/100(1/20) = 1/10995116000000
or 9.0949472e-14% (lol or 0.000000000000090949472%)
The rate of getting a Shiny Pokemon with all Max IVs and Hidden Ability
(with 3 stars of Potential, and Lv 100 search level, and using the maximum probability of getting Shiny based on the test 1/308)
1/308 1/32768 23/100 = 23/1009254400
or 2.27891005e-8% (0.00000002.27891005%)
The rate of getting a Shiny Pokemon with all Max IVs and Hidden Ability is within 1/10995116000000 and 23/1009254400
The rate of Hidden Abilities taken from Bulbapedia are not exact values, but based on observation.
The rate of Shiny rate of the DexNav was based on a research conducted by /u/Galactiiiic and others on Reddit are not exact values, but based on their test.
*All the variables factored in are without the use of Shiny Charm.
Sources:
http://bulbapedia.bulbagarden.net/wiki/Shiny_Pokémon#Generation_VI
http://bulbapedia.bulbagarden.net/wiki/DexNav
Shiny Encounter data provided by/u/Galactiiiic of Reddit.
https://docs.google.com/spreadsheets/d/1wIMVoliCKs4HkDvpgFkPfP5HKxZb0De3cFQidBoYB1c/edit#gid=0
https://docs.google.com/document/d/1e4y6TW21TcUSG_Qe2sKzXoMgkwttLlclzV94_DBpUgU/edit
[1]: https://docs.google.com/document/d/1e4y6TW21TcUSG_Qe2sKzXoMgkwttLlclzV94_DBpUgU/edit
Edit: Fixed 5 IVs to 6 IVs
