Can Pokemon evolve while walking with you?

Not really. To the best of my knowledge, the only way for walking to have any impact on the level of a Pokemon is to either have an egg in your party or have one of your Pokemon in the day care center at the time you're doing the walking.

A regular Pokemon will never evolve when you walk, it will always level up at the end of the battle if it leveled up to the appropriate level during said battle, although using a stone is the only way to evolve a Pokemon outside of battle, just find a stone and a Pokemon that are compatible (like Eevee and a thunderstone for Jolteon) go into the bag, go under items, find the stone (thunderstone, fire stone, water stone, dark stone, sun stone, light stone, leaf stone) select it and chose use, then when a list of all your Pokemon currently in your party, some may say unable, ingnore these if you get one thats not you can evolve it by selecting it, you will then see an evolution screen and then you will have the evolved Pokemon, enjoy!

The only two ways for a Pokemon to evolve while walking are when you put one or two in the daycare or if you have an egg with you. Yes, the Pokemon that evolve with happiness(and the Pokemon to no effect of happiness)will get happier if you walk with them. After a while of this, you should level it up one and it should evolve. But the Pokemon to no effect of happiness will not evolve this way. The egg and Pokemon in the daycare will eventually evolve. The time it takes for an egg to hatch depends on the Pokemon, as Professor Elm has said over the phone, and for every step the Pokemon in the daycare will gain 1 Exp. Point.

So yes, it should evolve if it is a Pokemon to effects of happiness, an egg, or a Pokemon in the daycare.

Hope this helps!

~PoKeMaStEr3591