Something that occurred to me: Clinton needing to have his hand mount removed would be a major operation. It appears to be pretty firmly integrated into his forearm - it probably attaches to the bone and is controlled by the nerves of his arm. So removing it would be tantamount to another amputation.
In short, they'd probably try to find any possible alternative to an MRI that they could in Clinton's case.
I'm not quite sure it would be as drastic as that.
A lot of current medical implants have to be MRI safe and are made from non-ferromagnetic materials. These materials would be tested to ensure that they can be used for implantation, but also so that they can be used and examined with the minimum of invasive procedures. Often in an emergency time is critical and if you have remove an implant because you think it might interfere with equipment, that's time that is potentially fatal for a patient.
Just a quick word - every implant is stringently tested because it is used on a patient, it will be used on a person. Every piece has a limit and those limits are generally within the range that is healthy for the patient. Because an MRI could last a few minutes to an hour, some implants can be made to last several hours under the conditions of an MRI, but after that, it will fail.
It also has to come down with the biocompatibility of the materials used, for example titanium is used a lot in hip and knee replacements not only because its got a strength similar to bone, but because it also does not provoke an immune response when implanted. The last thing you want is the body rejecting an implant. Because titanium has an extremely weak magnetic field compared to ferromagnetic materials, its used more often because it is safer from the patient's standpoint and because it can be safely checked through x-rays and MRIs.
With regards to Clinton, I don't think his prosthetic mounting would be permanently attached. If you have something with working parts close to human flesh, it can cause irritation or in rarer instances infection. So it would be case that if Clinton is feeling something on his stump, a doctor would need to examine the stump, which would necessitate removing the mounting.
Likewise, depending on how old Clinton was when he lost his hand, either he would have had to wait until he was fully grown before getting the prosthesis or gone through a series of prostheses as he was growing up. The hand he has is possible myoelectric, in that it is powered and controlled by the faint bioelectric that all muscles have, picked up by electrodes implanted in the arm.
The fact that Claire said that Clinton has to have the mounting removed, probably tells that there is a lock or method to quickly and safely take it off.