Equilibrium means to make the system stay at rest. i.e., to make it stay as it is.
The way you have defined the forces to attain equilibrium is correct (as in the image) as you can attain both linear and angular equilibrium.
Linear equilibrium is obtained if the sum of the forces( i.e., the net force ) in each direction is zero. And angular equilibrium, if the net torque is zero.
But, taking $ Ray $ and $ Rcy $ to be in opposite direction can assume linear equilibrium if :
1) $ Ray + 10= Rcy $ if $ Ray $ is downwards and $ Rcy $ is upwards.
2) $ Rcy + 10 = Ray $ if its vice versa.
But, both the above two cases won't satisfy angular equilibrium as the net torque won't be zero.
Taking both forces to be downwards won't satisfy both equilibriums.
So, you have to define your forces in such a way that they ensure the system to attain overall equilibrium.
So, the only way is as you have taken in the image above.
Of course you will get different values as you change the direction of each force as your main concern is to sustain equilibrium and the net force in one direction has to be canceled out by that force that is in the opposite direction.
" $ Ray $ is pointing down and $ Rcy $ is pointing up, it changes the answer for
$ Ray $."
This is because, in the earlier case the 10$ N $ force was cancelled out by $ Ray $ and $ Rcy $ together but when $ Ray $ is pointing down and $ Rcy $ is pointing up the net force of 10$ N $ and $ Ray $ has to be cancelled out by $ Rcy $ alone and hence $ Rcy $ has to be greater in magnitude $.