I don't see how the doctor could determine that you were born with a herniated disc, unless you had imaging of your back done as an infant. Kind of makes me shake my head if the doctor did indeed say that. I'd be more likely to believe that due to poor lifting mechanics, posture, daily activity, etc that the disc started to bulge slightly. The fact that your father had a bulged disc really has no bearing on your back, other than the possibility that you have similar limb length which creates poor levers for certain lifts and places more stress on your back.
Also, did you only receive an XRay or was an MRI taken as well? If just an XRay was given, than AT BEST, the Xray will show decreased disc height which could mean disc degeneration occurring. An xray is not suited to show herniation/bulging of the discs; that's what a MRI is for.
x2 for what beefcake said. Disc bulge results on imaging doesn't mean anything unless symptoms are following. And as of right now, you are saying you don't have any low back pain nor any referral leg pain, and the only time you seem to have the pain is when the load is higher during squats, which IMO, shows that your body is not ready to handle those higher loads (mobility issue, stability issue, technique issue, etc), which may or may not also be impacted by some disc degeneration.