Shield: This value represents the reduction of damage that Sentry receives while the shield is active. At base, the amount of damage you receive while the shield is active will be reduced by 20%.
Shield Health: The maximum amount of (pre-reduction) damage you can take before the shield ceases protecting you. A shield health of 600 with a Shield value of 20% prevents a maximum of 120 damage (20% of 600). Any damage taken in excess of your Shield Health will not be reduced, nor will it add to BHS's damage.
Damage Conversion: What percentage of the damage taken (before being reduced) gets converted into damage added to the attack portion of BHS. So with the 65% Damage Conversion, if you take an attack that would have dealt 500, your BHS attack power gets increased by 325.
Note: As explained by one of the devs here, damage received from turrets and the flame trap on AI Station 205 only counts half for increasing your damage.
Minimum Damage and Maximum Damage: These refer to the range of damage that can be dealt by the attack portion of BHS. The Minimum Damage value is the amount it will deal if your shield absorbs no damage, while the Maximum value is the amount it will deal if your shield absorbs the most damage it can. As Maximum is a product of your Shield Health and Damage Conversion, that entry is simply for informational purposes.
There's no complicated formula involved, the only complication is that damage stops being reduced once you exceed the Shield Health value. If you take less damage than that, you take (Shield%) less damage and BHS's attack power is increased by (Damage Conversion%) of the damage you would have taken. Any damage taken in excess of Shield Health is not affected by either Shield or Damage Conversion.
Example: You have BHS with the stats mentioned in the question. You're hit by an attack that would deal 500 damage. Your Shield Health is 600, so it can help with the whole attack. The damage is reduced by 20%, meaning Sentry takes 400. 65% of the original 500 damage (325) is added to the damage of your next BHS attack, raising its damage from 150 to 475.
Example 2: You have BHS with the stats mentioned in the question. You're hit by an attack that would deal 1,000 damage. Your Shield Health is 600, so only that much of the attack is affected by the shield, reducing it to 480, plus the remaining 400 that wasn't protected against, for a total of 880 damage that Sentry receives. 65% of the 600 damage the shield took (390) is added to the damage of your next BHS attack, raising it from 150 to 540 (the Maximum Damage).