Is there a circuit like a flip flop but with 1 input and 4 outputs
This can be done with a shift register.  You shift a 1 into the first bit, then 0 after that.

However, there is a big gotcha with what you describe.  Switches bounce.  Your pushbutton may actually go thru dozens of on/off transitions during the first few ms of either edge.

Since you really need to do debouncing too, what you want is most easily handled in a microcontroller.  It can do the debouncing, then activate multiple outputs in any way you choose based on debounced switch transistions.