Can flooding cause the Drains in your house to back up?

The quick answer to this question is, No

However it depends on your set up.

If you are connected to a sewer then you are safe from back-ups in any weather condition.

If you have a septic tank (that does not work well to begin with) you may experience a backup issue as the additional water in the ground may get into the tank.

But if your tank is generally Ok, then your Ok.