How do I provide a negative current?
If the LEDs turn on when you apply 5V to ground then what you have identified as GND on the board is actually VCC.
There is no such thing as negative current: current is either sourced or sinked, depending if it’s exiting or entering into a node.

The diodes (LED) are one way valves, so you have a clear current direction: once that is established, identifying a button press can work either triggering when it gets LOW or when it gets HIGH, in other words that can be easily reversed.

If previously you where detecting a button press with digitalRead(BTN_PIN) == HIGH then just switch to digitalRead(BTN_PIN) == LOW and you have your solution.