Bit sequence from USB HID Joystick
But I'm unable to find what bit sequence is sent when a button is pressed on the joystick. I would like to find what that sequence is.


There isn't one.

USB devices operate on a polling basis. They do not communicate over the bus outside of a transaction from the host. Getting to the point of communicating with the device takes a significant amount of work to enumerate the device and configure it with an address.

Even once that's all done, there isn't a specific message used when a button is pressed. The status report response from the device will have a field which includes a bitmap representing the buttons on the mouse -- one of the bits in that field will be set when the button is being held down, and clear when it is not.

Since the Arduino lacks a USB host peripheral, and runs at a relatively low speed (8-16 MHz) compared to USB line rate (1.5 or 12 Mbit/sec), it will be excruciatingly difficult, if not impossible, to implement a USB host on an Arduino device. Use an accessory which implements a USB host for you, like the USB Host Shield, or use a different microcontroller which supports USB host operation.