When writing code like this in C++:
bool allTrue = true;
allTrue = allTrue && check_foo();
allTrue = allTrue && check_bar();
check_bar() will not be evaluated if check_foo() returned false. This is called short-circuiting or short-circuit evaluation and is part of the lazy evaluation principle.
Does this work with the compound assignment operator &=?
bool allTrue = true;
allTrue &= check_foo();
allTrue &= check_bar(); //what now?
For logical OR replace all & with | and true with false.
From C++11 5.17 Assignment and compound assignment operators:
The behavior of an expression of the form E1 op = E2 is equivalent to E1 = E1 op E2 except that E1 is evaluated only once.
However, you're mixing up logical AND which does short-circuit, and the bitwise AND which never does.
The text snippet &&=, which would be how you would do what you're asking about, is nowhere to be found in the standard. The reason for that is that it doesn't actually exist: there is no logical-and-assignment operator.