One approach for eagle pose arms is to cross the arms in front of you with elbows bent and then grab the back of the opposite shoulder with both hands.
You can then use your hands to pull your elbows past each other. You can hold this for a few breaths.
Then try to rotate your upper arms outwards so that your forearms cross. As you do this you might find it helpful to spread your shoulder blades (middle and right below).
Once your forearms cross you can then hook the fingers of your lower arm into the palm of your upper arm.
Once your arms are in eagle pose you can vary the shoulder stretch by reaching your forearms forwards (below left) or lifting your elbows (below center).
You can also tilt your arms to upper arm side to give the upper arm an external rotation stretch (below right). And depending on how you position your shoulders you can also tilt your arms to the lower arm side. (You might find it helpful to bend your spine to the lower arm side also when doing this.)
Another option for getting the arms into eagle pose arms is to reach both arms out to the sides with the arm that is going to be on top slightly higher.
Keeping your elbows straight, cross your arms in front of your body. As your arms cross, bend your elbows and externally rotate both upper arms to move your forearms past each other. You can then place the fingers of your lower arm to the inside of the wrist of your upper arm. If at first you don't succeed you can repeat this exercise slowly and smoothly with your breath. Inhale arms apart, exhale and cross the arms and externally rotate.
Once you can bind your arms in eagle pose you can also practice the leg position separately and then put the two together.
To make binding the legs easier it helps to roll the thighs inwards. Another action that you can practice is lifting the hip of the top leg. With one leg on top of the other (but not bound) you can practice lifting and lowering the hip of the upper leg relative to the hip of the lower leg.
It may help to press the top leg down against the thigh of the lower leg.
Next practice rotating one leg inwards while your weight is over the other leg. In the pictures below my weight is on my left foot and I'm practicing internally rotating the right leg.
1. Lifting the Hip. 2. Internally Rotating the Thigh
Try the same action with one leg on top of the other. Lift the hip of the upper leg and as you do so roll the thigh of that legs inwards. You may find that as you do so the foot of that leg naturally moves towards the ankle of the standing shin.
Note that in this position both knees are bent. You can vary the degree of knee bend to find a position that makes it easier to eventually bind the feet.
Even if you can't bind I'd suggest not worrying about it too much. You can get some strengthening benefits from the pose just by trying to keep the upper leg foot close to the lower leg ankle.
One of the tricks with Eagle pose (like Tree pose) is staying balanced when pressing one thigh against the other. Eagle pose is extra tricky perhaps because the legs are crossed. And so to overcome balance difficulties stay focused on the supporting foot even as you lift one hip and internally rotate the thigh.
When moving into Eagle pose you can do the arms first and then try to cross the legs. I tend to cross the legs first and then do the arms.
Another challenge is to cross both simultaneously.
As to which way to cross the arms relative to the legs you can have the right leg and right arm both on top or right leg on top with left arm on top. Both with have slightly different effects and rather than saying which way is the right way to do Eagle pose I'd suggest experimenting with both.