Good question, and as fundamental a question as anyone could ask. I think that the laws of logic must be not only independent of human minds but independent of any minds, including God's mind if such exists. At any rate, I don't think anyone can see how it could be otherwise.
To say that the laws of logic depend on human or divine minds is to imply that the following conditional statement is nontrivially true:
If (1) human or divine minds had been different enough, then (2) all of the laws of logic would be different from what they are.
(By "nontrivially true," I mean that the statement is true not merely on the ground that (1), its antecedent, is logically impossible. If (1) is logically impossible, then the conditional statement is trivially true, even if (2), its consequent, is also logically impossible.)
We can't make sense of the italicized statement without presupposing that (2) is false. If the italicized statement means anything, then it doesn't mean this: If (1) human or divine minds had been different enough, then (~ 2) not all of the laws of logic would be different from what they are. But, of course, my assertion just now about the statement's meaning itself depends on holding fixed at least some of the laws of logic, i.e., it depends on presupposing (~ 2) even on the assumption that (1) is true. Therefore, we understand the italicized statement only if we presuppose that it can't be nontrivially true.
As for the nonphysical existence of the laws of logic, you might look at what I wrote in reply to Question 24874.