String Comparison - Not Equal To problem in While Loop

As PaulS stated in the future please post the complete sketch after applying formatting. But assuming that 'button' is a global declared as …

char*   buuton;

… your code should be …

    clearScreen ();
    lcd.print("Setup Menu - Active");
    lcd.setCursor(0, 2);
    lcd.print("Pulse Count: ");
    lcd.setCursor(13, 2);
    lcd.print(pulseCount);

    do 
    {
        increment(pulseCount, 1);

        lcd.setCursor(13, 2);
        lcd.print(pulseCount);

        Serial.print(button);  //Debug Code
    } while (button != "Accept" || button != "Reject");
    
    
    if ( 0 == strcmp(button, "Reject") )
    {
        menuSetup1();
    }

    lcd.setCursor(0, 3);
    lcd.print("Trigger Set to: ");
    lcd.setCursor(16, 3);
    lcd.print(pulseCount);

    buttonPress();
    if ( 0 == strcmp(button, "Reject") )
    {
        menuTop();
    }
}    

int increment(int startValue, int value)
{
    buttonPress();
    if ( 0 == strcmp(button, "Up") )
    {
        pulseCount = pulseCount + value;
    }
    else if ( 0 == strcmp(button, "Down") )
    {
        pulseCount = pulseCount - value;
    }
    else
    {
        return pulseCount;
    }
}