In breeding only males can pass down moves to the egg, and females determine what Pokemon it will be. To breed two Pokemon they must have at least one matching egg group. You may already know this so I will cut to the chase, and tell you how to get both moves onto a Squirtle.
The easiest way to get both of these moves on a Squirtle is to breed a female Squirtle (Wartortle or Blastiose) with a male Clawitzer who learns both Aura Sphere and Dragon Pulse by leveling up. Squirtle (and its evolutions) have two egg groups: Monster and Water 1, and Clawitzer has the egg groups Water 1, and Water 3. Since Squirtle and Clawitzer have at least one matching egg group they can breed to make eggs.
My source for my answer is this very site!
It's basically all mix and match trying to find a way to get the moves you want. I spent two days trying to figure out how to get Stealth Rock and Curse onto a Larvitar, until I realized I could breed curse onto a male Tyrunt with a Slowpoke, and level up the Tyrunt until it learned Stealth rock, and breed it with a Larvitar and viola! It may take a while but you will get the hang of it, however complicated it is.