Cannot break while loop in C++

while (run = true) is always true, and it has the side effect of setting run to true.

Did you want ==? Didn't your compiler warn you of this? Better still, drop the tautology run == true and write while (run).

Then fix the other conditional to while (check != 'y'&& check != 'n'). Think about this: check != 'y'|| check != 'n' is always true (if check is y, say, then it's therefore not n). That logic error is extremely common.

Finally, I'm not sure what you really achieve with the introduction and therefore the maintenance of run. Why not replace it with while (true), and write return 0; in place of break?

Here is how it should be done:

do {
    cout << "Do you want to try again?(y/n)\n"; cin >> check;
    cin.clear();
} while (check != 'y' && check !='n');

Full code:

#include <iostream>
using namespace std;

int main()
{
    bool run = true;
    while (run) {
        double len, width, area, volume, height;
        char check;
        cout << "Please, enter length,width and height of a prism(rectangular)!";
        cin >> len >> width >> height;
        area = width*len;
        volume = area*height;
        cout << "The area of rectangle is equal to: " << area << "\n" << "The volume of rectangular prism is equal to: " << volume << "\n";
        do {
            cout << "Do you want to try again?(y/n)\n"; cin >> check;
            cin.clear();
        } while (check != 'y' && check !='n');

        if (check == 'n') {
            run = false;
        }
    }
    return 0;
}

check != 'y' || check != 'n'

This is always true, since check cannot equal to 'y' and 'n' at the same time. You should change it to

check != 'y' && check != 'n'

By the way, you wrote at the outer loop

while (run = true)

This will result in run getting assigned to true at the begining of each outer loop and since this expression also returns true, you end up in a dead loop. This should be changed to

while (run == true)

Enabling compiler warning and possibly -Werror will help avoid such problem. And of course for bool variables you can just write while (run).

For starters the variable run in this loop

bool run = true;
while(run = true){
    // ...
    if(check == 'n'){
        run = false;
        break;
    }
}

is redundant. In fact the loop does not depend on the variable run. The loop stops its iteration due to the break statement in the if statement

    if(check == 'n'){
        run = false;
        break;
    }

independent on whether the Boolean literal true will be assigned to the variable as it is done in the condition of the while statement (it seems you mean comparison run == true instead of the assignment)

while(run = true){

or the Boolean literal false will be assigned to the variable as it is done in the if statement

    if(check == 'n'){
        run = false;
        break;
    }

In fact you have an infinite loop that can be stopped only by the break statement.

So just remove the variable and write the loop the following way

while( true ){
    // ...
    if(check == 'n'){
        break;
    }
}

As for the second loop

    while(check != 'y'|| check != 'n'){
        cout <<"Please enter y or n\n";
        cin >> check;
    }

then its condition will always yield true because neither character can be simultaneously equal to 'y' and to 'n'.

So a valid condition should look like

    while(check != 'y' && check != 'n'){
        cout <<"Please enter y or n\n";
        cin >> check;
    }

Or you could make it even more readable the following way

    while( not ( check == 'y' || check == 'n' ) ){
        cout <<"Please enter y or n\n";
        cin >> check;
    }

#include <iostream>
using namespace std;

int main()
{
bool run = true;
while(run == true){
    double len,width,area,volume,height;
    char check;
    cout << "Please, enter length,width and height of a prism(rectangular)!";
    cin >> len >> width >> height;
    area = width*len;
    volume = area*height;
    cout << "The area of rectangle is equal to: " << area << "\n" << "The volume of rectangular prism is equal to: " << volume << "\n";
    cout << "Do you want to try again?(y/n)\n";
    cin >> check;
    while(check != 'y' && check != 'n'){
        cout <<"Please enter y or n\n";
        cin >> check;
    }
    if(check == 'n'){
            run = false;
            break;
            }
    }
    return 0;
}

You must change the = sign in the first loop for == and also need &&instead of ||in the other while loop.

Try having check as a string instead of a char.