Pokemon-go – Does evolving your buddy retain the amount of walking you have done thus far

Yes, you still need to feed your Eevee candies to evolve it. The trick you're referring to is a popular trick to control which Pokémon Eevee will evolve into.

In the traditional games, you need to give Eevee an evolution stone to make it evolve. A Water Stone for Vaporeon, a Thunder Stone for Jolteon, and a Fire Stone for Flareon. Since these stones don't exist in Pokémon Go, Eevee evolves randomly into either.

To avoid having to rely on luck, people have attempted to determine what Eevee will evolve into. Eventually, it was discovered that renaming Eevee allows players to define which Pokémon Eevee will evolve into, assuming that evolution had not been registered in their Pokédex before:

Restarting the game is only used to ensure that your Eevee has been renamed, since the servers aren't always available. By restarting, you can determine if your Eevee has been successfully renamed before you spend candies to evolve it.

Of course, this doesn't change the fact that you still need to candies to evolve your Pokémon. Also keep in mind that the trick only works if you haven't registered the respective evolution in your Pokédex yet.

According to this report,

Buddy Pokémon can be swapped out at will in “Pokémon Go’s” latest update, but doing so has no impact on your total kilometers walked stat. It will, however, reset progress toward your next candy. Just make sure all incoming candies are earned before making a switch.

So, the answer is yes, but the progress towards your reward will be reset each time you switch.

To change your buddy, follow these steps:

Click the button on the bottom left of the screen showing your character’s face.

Hit the button on the bottom right of the screen with the three lines.

Select “buddy.”

Hit the button on the bottom right of the screen with the two arrows.

You will be asked “Are you sure you want to swap your buddy?” Hit yes.

Select the Pokemon you want to have as your new buddy.