No, it doesn't stay the same. When a Pokemon is hatched by breeding it's nature and IV's will be randomly assigned, just like if you caught a new one. So there is a good chance that it will be different then the original, but no guarantee there.
In case you wondered, if you let a Pokemon hold an everstone (male or female) then there is a 50% chance of the newly hatched having the same nature. Just wanted to share :}