Yes, this form of initialization is supported by all C++ compilers. It is a part of C++ language. In fact, it is an idiom that came to C++ from C language. In C language = { 0 } is an idiomatic universal zero-initializer. This is also almost the case in C++.
Since this initalizer is universal, for bool array you don't really need a different "syntax". 0 works as an initializer for bool type as well, so
bool myBoolArray[ARRAY_SIZE] = { 0 };
is guaranteed to initialize the entire array with false. As well as
char* myPtrArray[ARRAY_SIZE] = { 0 };
in guaranteed to initialize the whole array with null-pointers of type char *.
If you believe it improves readability, you can certainly use
bool myBoolArray[ARRAY_SIZE] = { false };
char* myPtrArray[ARRAY_SIZE] = { nullptr };
but the point is that = { 0 } variant gives you exactly the same result.
However, in C++ = { 0 } might not work for all types, like enum types, for example, which cannot be initialized with integral 0. But C++ supports the shorter form
T myArray[ARRAY_SIZE] = {};
i.e. just an empty pair of {}. This will default-initialize an array of any type (assuming the elements allow default initialization), which means that for basic (scalar) types the entire array will be properly zero-initialized.