You can try the below using a wire between pin 4 and ground to simulate your button (wire connected, servos move, wire not connected, servos don't move). I haven't tried this so YMMV.
#include <Servo.h>
int button = 4; //button pin, connect to ground to move servo
int press1 = 0;
Servo panServo; // Create a servo object for the pan (horizontal) servo
Servo tiltServo; // Create a servo object for the tilt (vertical) servo
Servo zoomServo; // Create a servo object for zoom (left/right) servo
Servo downServo; // Create a servo obhect for record (down/up) servo
int panPin = 0; // Analog input for the joystick pan axis
int tiltPin = 1; // Analog input for the joystick tiltl axis
int zoomPin = 2; // Analog input for the zoom joystick axis
int downPin = 3; // Analog input for the down axis
int tiltVal; // Value read from the vertical axis
int panVal; // Value read from the horizontal axis
int zoomVal; // Value read from the zoom axis
int downVal; // Value read from the down axis
void setup()
{
pinMode(button, INPUT); //arduino monitor pin state
digitalWrite(4, HIGH); //enable pullups to make pin high
panServo.attach(9); // use pin 9 PWM output for pan servo
tiltServo.attach(10); // use pin 10 PWM output for tilt servo
zoomServo.attach(6); // use pin 6 PWM output for zoom servo
downServo.attach(5); // use pin 5 PWM output for down servo
}
void loop ()
{
panVal = analogRead(panPin); // read pan joystick position
panVal = map(panVal, 0, 1023, 0, 179); // scale reading to servo
if (press1 == LOW) panServo.write(panVal); //move servo to required position
tiltVal = analogRead(tiltPin); // read pan joystick position
tiltVal = map(tiltVal, 0, 1023, 0, 179); // scale reading to servo
if (press1 == LOW) tiltServo.write(tiltVal); //move servo to required postion
zoomVal = analogRead(zoomPin); // read zoom joystick position
zoomVal = map(zoomVal, 0, 1023, 0, 179); // scale reading to servo
if (press1 == LOW) zoomServo.write(zoomVal); //move servo to required position
downVal = analogRead(downPin); // read record joystick position
downVal = map(downVal, 0, 1023, 0, 179); // scale reading to servo
if (press1 == LOW) downServo.write(downVal); //move servo to required position
}