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## [Rust](https://www.rust-lang.org/), 236 bytes

```rust
|i:u64|(i|(i&71209857637481724)<<6|(i&280371153272574)<<15|(i&140185576636287)<<17|(i&17802464409370431)<<10|(i&4557430888798830336)>>6|(i&9187201950435704832)>>15|(i&18374403900871409664)>>17|(i&18229723555195321344)>>10).count_zeros()
```

[All test cases on Rust Playground](https://play.rust-lang.org/?version=stable&mode=debug&edition=2021&gist=d3bbad94c532de2c8209dcd7ce193459)

Takes input as a 64 bit unsigned integer where each 1-bit represents a knight and each 0-bit represents an empty square.

### Ungolfed version

```rust
let knight_safe_squares = |input_integer: u64| {
    // Moves in direction (2, -1)
    let mask = u64::from_str_radix("\
        00000000\
        11111100\
        11111100\
        11111100\
        11111100\
        11111100\
        11111100\
        11111100\
    ", 2).unwrap();
    let pieces_to_move = input_integer & mask;
    let moved1 = pieces_to_move << 6;

    // Moves in direction (1, -2)
    let mask = u64::from_str_radix("\
        00000000\
        00000000\
        11111110\
        11111110\
        11111110\
        11111110\
        11111110\
        11111110\
    ", 2).unwrap();
    let pieces_to_move = input_integer & mask;
    let moved2 = pieces_to_move << 15;

    // Moves in direction (-1, -2)
    let mask = u64::from_str_radix("\
        00000000\
        00000000\
        01111111\
        01111111\
        01111111\
        01111111\
        01111111\
        01111111\
    ", 2).unwrap();
    let pieces_to_move = input_integer & mask;
    let moved3 = pieces_to_move << 17;

    // Moves in direction (-2, -1)
    let mask = u64::from_str_radix("\
        00000000\
        00111111\
        00111111\
        00111111\
        00111111\
        00111111\
        00111111\
        00111111\
    ", 2).unwrap();
    let pieces_to_move = input_integer & mask;
    let moved4 = pieces_to_move << 10;

    // Moves in direction (-2, 1)
    let mask = u64::from_str_radix("\
        00111111\
        00111111\
        00111111\
        00111111\
        00111111\
        00111111\
        00111111\
        00000000\
    ", 2).unwrap();
    let pieces_to_move = input_integer & mask;
    let moved5 = pieces_to_move >> 6;

    // Moves in direction (-1, 2)
    let mask = u64::from_str_radix("\
        01111111\
        01111111\
        01111111\
        01111111\
        01111111\
        01111111\
        00000000\
        00000000\
    ", 2).unwrap();
    let pieces_to_move = input_integer & mask;
    let moved6 = pieces_to_move >> 15;

    // Moves in direction (1, 2)
    let mask = u64::from_str_radix("\
        11111110\
        11111110\
        11111110\
        11111110\
        11111110\
        11111110\
        00000000\
        00000000\
    ", 2).unwrap();
    let pieces_to_move = input_integer & mask;
    let moved7 = pieces_to_move >> 17;

    // Moves in direction (2, 1)
    let mask = u64::from_str_radix("\
        11111100\
        11111100\
        11111100\
        11111100\
        11111100\
        11111100\
        11111100\
        00000000\
    ", 2).unwrap();
    let pieces_to_move = input_integer & mask;
    let moved8 = pieces_to_move >> 10;

    (input_integer | moved1 | moved2 | moved3 | moved4 | moved5 | moved6 | moved7 | moved8).count_zeros()
};
```

### Explanation
For each of the 8 directions in which a knight can potentially move, the input integer is bit shifted by the appropriate number of places to take each bit to the destination position of a knight moving in that particular direction. This would take some knights wrapping over the edge of the board, and others off the board altogether. These knights are first removed by a bitwise AND with a mask that has 0-bits at these problematic positions. There is a different mask for each direction, each with 0-bits for the knights that are not permitted to move in that direction.

Now for each of the 8 directions in which a knight can potentially move, these bit shifted results (`moved1` to `moved8` in the ungolfed version) has 1-bits in each of the destination positions that  the knights in the input would end up in the respective direction.

The bitwise OR of the input (the starting positions of the knights) with all 8 of the potential moved positions leaves a 1-bit in every position that is either a knight or the result of a legal move for one of those knights.

The output is the number of 0-bits in this final bit string.

In the golfed version, the masks are the equivalent 64 bit unsigned integers, instead of bit strings.