Let's analyze each option step by step:

Option A:
- If events $A$ and $B$ are mutually exclusive, it means $P(A \cap B) = 0$.
- However, $A$ and $\overline{B}$ being mutually exclusive would imply $P(A \cap \overline{B}) = 0$. This is not necessarily true. For instance, if $A$ and $B$ are complementary, then $A = \overline{B}$, and they are not mutually exclusive.
- Therefore, option A is incorrect.

Option B:
- Independence between events $A$ and $B$ means $P(A \cap B) = P(A)P(B)$.
- For $\overline{A}$ and $\overline{B}$ to be independent, we need $P(\overline{A} \cap \overline{B}) = P(\overline{A})P(\overline{B})$.
- The independence of $A$ and $B$ implies the independence of $\overline{A}$ and $\overline{B}$, as well as $A$ and $\overline{B}$, and $\overline{A}$ and $B$.
- Therefore, option B is correct.

Option C:
- Given $A$ and $B$ are independent, $P(A \cap B) = P(A)P(B) = 0.6 \times 0.2 = 0.12$.
- The formula for the probability of $A$ or $B$ occurring is $P(A + B) = P(A) + P(B) - P(A \cap B) = 0.6 + 0.2 - 0.12 = 0.68$.
- Therefore, option C is incorrect because it states $P(A + B) = 0.8$.

Option D:
- For independent events $A$ and $B$, $P(A\overline{B}) = P(A)P(\overline{B})$.
- $P(\overline{B}) = 1 - P(B) = 1 - 0.7 = 0.3$.
- Thus, $P(A\overline{B}) = 0.8 \times 0.3 = 0.24$.
- Therefore, option D is correct.

In conclusion, the correct options are $\boxed{B \text{ and } D}$.