When a pushbutton and sensor are in a certain state, do something for a certain period of time (15 seconds)
Here below is modified code based on your requirement. There are some changes in your code and I also comment it out. I hope it will help you.    

/*
 6 Flashing LED Lights With A Pushbutton and FSR Sensor
*/

const int buttonPin = 8;

const int LedPin1 = 7;
const int LedPin2 = 6;
const int LedPin3 = 5;
const int LedPin4 = 4;
const int LedPin5 = 3;
const int LedPin6 = 2;

const int kDelay = 1000; // Take one varible "kDelay" and value = 1000

int buttonState = 0;
int SensorPin = A0;
int SensorPinReading;


void setup()
{
  Serial.begin(9600);
  pinMode(LedPin1, OUTPUT);
  pinMode(LedPin2, OUTPUT);
  pinMode(LedPin3, OUTPUT);
  pinMode(LedPin4, OUTPUT);
  pinMode(LedPin5, OUTPUT);
  pinMode(LedPin6, OUTPUT);

  pinMode(buttonPin,INPUT_PULLUP); // Provide PULLUP
  pinMode(SensorPin, INPUT);
}


void loop()
{
  SensorPinReading = analogRead(SensorPin);
  Serial.print("Analog reading = ");
  Serial.print(SensorPinReading);

  if(SensorPinReading &lt; 10)
  {
    Serial.println(" - No Pressure");
  }
  else if (SensorPinReading &lt; 200)
  {
    Serial.println(" - Light Touch");
  }
  else if (SensorPinReading &lt; 500)
  {
    Serial.println(" - Light Squeeze");
  }
  else if (SensorPinReading &lt; 800)
  {
    Serial.println(" - Medium Squeeze");
  }
  else
  {
    Serial.println(" - Big Squeeze");
  }


  buttonState = digitalRead(buttonPin);

  if(buttonState == HIGH &amp;&amp; SensorPinReading &gt; 10)
  {
    digitalWrite(LedPin1, HIGH);
    digitalWrite(LedPin2, HIGH);
    digitalWrite(LedPin3, HIGH);
    digitalWrite(LedPin4, HIGH);
    digitalWrite(LedPin5, HIGH);
    digitalWrite(LedPin6, HIGH);     
    delay(15 * kDelay);            // You need 15 Sec, multiply kDelay with 15                      
    digitalWrite(LedPin1, LOW);
    digitalWrite(LedPin2, LOW);
    digitalWrite(LedPin3, LOW);
    digitalWrite(LedPin4, LOW);
    digitalWrite(LedPin5, LOW);
    digitalWrite(LedPin6, LOW);                              
  }
  else
  {
    digitalWrite(LedPin1, LOW);
    digitalWrite(LedPin2, LOW);
    digitalWrite(LedPin3, LOW);
    digitalWrite(LedPin4, LOW);
    digitalWrite(LedPin5, LOW);
    digitalWrite(LedPin6, LOW);
  }
}