I want to run 6 Stepper Motors type 28BYJ-48 simultaneously making 3 turns c/w
and stop. Later, an outside switch will reactivate the program and the Steppers will now make
4 turns cc/w and stop. The coding needs to be for ARDUINO UNO R3. Is this possible
to do?

Will the 6 steppers always be required to do the same thing as one another ?

In a Word YES.

I will just come down to pin count on the Arduino you use if you want to control each motor independently.
With an UNO I think the IO pin count is 20??

Looks like you may have to get clever with some sort of multiplexing or use some external BCD latch.

Unless you can simply have 2 wires per motor,
1 x direction control,
1 x enable/step.

Yep Bobs question is important.

With an UNO I think the IO pin count is 20??

Less 0 and 1 for Serial, and I tend never seldom to use 13 since it has some quirks, so 17.

Much better to use a Mega and then there will be no shortage of pins.

How many steps per second will be required taking all 6 motors together. A 16 MHz Arduino may struggle if a high-step rate is required.