How to mount butcher's block counter above washer/dryer?
What kind of ledge would I need? 1x4's mounted to studs?

1x4 or 2x4 for a ledger board (cleat) will work.
Yes you want them attached to at least two studs.
You do not even need to secure it to the cleats if you do not want to, It is very heavy, as long as it sized properly to not slide far enough side to side and slide off the cleat on the other end.  If you do secure it, bolts are not necessary, two L brackets on each side with small screws will be fine.

Would I need support in the middle, for a 6' span of 1.5&quot; thick
counter?

NO. A butcher block (real hardwood) spanning 6 feet will not need center support, you could put a cleat along the back wall if you want.

Does the counter need to clear the top of the washer/dryer by a
specific amount? The dryer is electric not gas.

There is no rule about this that  i am aware of, You may want it low enough that things can not be set on the appliances and fall off the back.  Conversely, you may want enough room to reach in and shut the water supply off in an emergency.