continue ALLWAYS Illegal in switch in JS but break works fine

continue has absolutely nothing to do with switches, not in Javascript and not in C++:

int main()
{
    int x = 5, y = 0;
    switch (x) {
        case 1:
            continue;
        case 2:
            break;
        default:
            y = 4;
    }
}

error: continue statement not within a loop

If you wish to break out of the case, use break; otherwise, allow the case to fall through:

switch ("B")
{
    case "A":
        break;
    case "B":
    case "C":
        break;
    default:
        break;
}

If you're looking for a shortcut to jump to the next case then, no, you can't do this.

switch ("B")
{
    case "A":
        break;
    case "B":
        if (something) {
           continue; // nope, sorry, can't do this; use an else
        }

        // lots of code
    case "C":
        break;
    default:
        break;
}

I believe you can emulate what you want by using a labeled infinite loop:

var a = "B";
loop: while( true ) {
    switch (a)
    {
    case "A":
        break loop;
    case "B":
        a = "C";
        continue loop;
    case "C":
        break loop;
    default:
        break loop;
    }
}

Otherwise you should consider expressing what you want in some other way.

Even if you can pull this off, it would be a huge WTF. Just use if else if.

I think what you meant is:

switch ("B")
{
    case "A":
        break;
    case "B":
    case "C":
        break;
    default:
        break;
}

There is no need for continue. When B comes, it will move on to C.