Left and right shifts are bitwise operations, not logical operations. QB doesn't distinguish between logical and bitwise operators when it comes to AND, OR, XOR, NOT, EQV and IMP, but they function as bitwise operators. They just happen to work correctly when used as logical operators.
As for NOT AND (NAND) and NOT OR (NOR), you can just use NOT(a AND b), assuming I'm understanding correctly. And if I'm correct about that, then the request for exclusive NOT OR should already be implemented. XOR and EQV have opposite truth tables, meaning that if you apply NOT operations to the truth table of XOR, you'll have EQV. NOT(a XOR b) = a XNOR b = a EQV b.
If I'm not correct, I don't understand what you mean by "NOT AND", etc. because it simply doesn't make sense to do "a NOT AND b". What is the NOT operator being applied to? Another way to look at it is: what is the other argument to the AND operation? NOT is a keyword, and it can't be used as an argument.