When to use the different formats of the right hand rule? The question:
A rectangular loop of wire attached to a mass is put in a magnetic field (directed into the page) such that the top half of it is in the magnetic field and the bottom half of it is out of it. What should the direction of the current in the wire be so that the magnetic force can balance the force of gravity?
My pattern of thought:
Of course, to be able to counteract the force of gravity, the force must be directed upwards. I used the right-hand rule where my thumb as the force, my index finger as the magnetic field, and my middle finger as the current and I found the current must be counterclockwise.
Now in the answer key, it is solved by using the index finger, ring finger, and middle finger as the current, magnetic field, and force respectively. However, in other questions, it uses the other format (switching the current and force). They give two different answers. When do I use each?
 A: I found an article from Khan Academy that explains it nicely: https://www.khanacademy.org/test-prep/mcat/physical-processes/magnetism-mcat/a/using-the-right-hand-rule
In summary
The right-hand rule (RHR) is used differently depending on the scenario you are in. The following should help:

When there is a moving charge flying through space:

Use this:


When there is current flowing via a straight wire, use this:


When there is a coil of wire:


In your case
In your case for the rectangular piece of wire, use the last one I showed: Where your main fingers are the current in the coil and your thumb is the magnetic field.
Like this:

Where the finger in the red circle is the single wire that you have for the rectangular wire. If the current is going the other way (clockwise) then just flip your hand upside down.
The answer
So yes, according to this specific RHR, the current should be in the counterclockwise direction and the magnetic field should point upwards and (given enough current) it can counteract gravity.
This RHR is the best one that should give you the right answer. Do not use the 3 finger arrangement for loops of wire, use the curled fingers instead (the last image above).

Personally, I use this when there is a charge floating via space:

I usually get confused with the 3 finger arrangement, so I use this instead.
