In short: no they do not reset.
But it's slightly more complicated because eggs don't hatch based on steps directly. Each egg has a number of "egg cycles" that must be completed. An egg cycle triggers every 257 steps, so every 257 steps that you walk around, the number of remaining egg cycles (for the eggs in your party) decreases by 1. When the egg cycles gets to 0 the egg hatches.
If you put an egg in the PC, the remaining number of egg cycles is not reset but it won't go down while it's in the PC. So the exact number of steps required could change slightly because you may be at a different point in the egg cycle when you withdraw it again.
Example: you get an egg with 15 egg cycles and you walk 257 steps. Its egg cycle will have gone down to 14. Then you walk 256 steps to the PC and deposit the egg. Then you walk 1 more step - now another egg cycle triggers, but the egg is in the PC so it stays at 14. Then you withdraw the egg and now you need to walk another 257 steps for it to go down to 13. Hope that makes sense!
