Perhaps the question should be why shouldn't they?
As a UK citizen I observe the Presidential elections with a great deal of interest. After all, the results will affect us all as global citizens, and indeed whatever the rules for the United States Presidential candidates, they should also apply to any other country that holds elections.
I believe it is essential that candidates should disclose the state of their health because they might be suffering from a condition that would have a direct bearing on whether they can function effectively at high level. This would not necessarily mean that every person in the world knows about every head cold or blemish about them. It should be possible to submit their health records for audit and clearance as a healthy candidate without those records becoming public knowledge.
The job of President is one of the greatest and most responsible positions in the world, and the candidates should have the same legal and moral obligation to submit the status of their health to scrutiny in the same way that any other working person at whatever level must do. There is a hierarchal scale of legislation in any government to examine personnel records for accuracy and credibility, and this should be used to determine the health risks of a candidate Employees do not have access to a colleague's medical records, and there is no reason why high level employees of the nation should either. Even people who work as volunteers have to be declared fit for purpose, so why should a presidential candidate have special privileges?
Health problems may not be such a concern if they are something that can be managed. Many people work in important positions with conditions such as diabetes or with artificial limbs. Even mental health problems can be controlled effectively so that the sufferer can work.
However, imagine a scenario where a candidate elected to President has not declared a health problem that later, when he is in office, affects his ability to remain functional. Imagine a President with an undeclared cancer, who needs chemotherapy to contain the disease. It is difficult enough to have to deal with such an illness let alone remain alert and in control while undertaking treatment. Worse still, imagine a President who was diagnosed with schizophrenia and was not required to declare it. Imagine the holocaust that could result from such an emission.
Everyone deserves a private life but when someone puts himself in the public eye to attain the highest possible position of power and authority, that person must be open to public scrutiny.
The Presidency is a tough job. It needs a tough and fit person at the top.
The term, "public scrutiny" is where the system could fall short. How would the general voting public know if the candidates' health records were properly inspected? Who could be trusted to do the job without prejudice? Are the records accurate? All these are right and proper questions that voters should ask. Getting satisfactory answers is quite another matter.
In business, if an employee fails to declare a medical condition on his application form, and is later found out because it has affected his performance; that employee could lose his job. This only creates minor problems that will be resolved when someone else is appointed to the position. A president is not so easily dismissed, and the ramifications could well be catastrophic. As a global player, the President must be in command of the situation at all times and if he is chronically ill this will not be the case. The hours are long, the stress horrendous, and the human body has limitations. If a president is not up to the job because of an undeclared illness, it could well lead to an international crisis. In addition, citizens could become disenchanted and disillusioned without an effective leader.
One clear example of this is the effect that Queen Victoria had on the people of the United Kingdom when she retreated into depression, mourning her husband's death for over ten years. She was criticised severely for her dereliction of duties and this led to the emergence of a strong republican movement that could well have destabilised the country. While this was not an illness that had been withheld from public knowledge prior to her becoming monarch, her ability to honour her future obligations was surely in question when she refused to communicate her condition to her people, and the effects were devastating. She could not easily be dismissed, and the problem could not be resolved for many years. Clearly this state of affairs could not have been foreseen, but this is precisely why candidates should openly declare any condition that might lead to greater problems when in office. It is too late once the President has been elected.
You might say that a candidate should not enter the Presidential race if he/she is not declared fit to run for office, but the egotistic nature of someone in that position might cloud his/her vision of what is important, and it is essential that a fair vetting process is implemented to eliminate problems.
So, how does one ensure that a candidate retains some personal privacy while satisfying the general public's demand for openness in all transactions? One answer might be to allow the opposing candidates to see the records and give them the right to challenge their opponent on health grounds without declaring the nature of the problem. Another may well be that citizens will accept that a panel of senior judges has examined the records and declared the candidates fit to stand for election. It may be enough that the administrative functions of the government will satisfy the public need to know that records are being checked. Of course, a candidate might decide that he/she has nothing to hide and would volunteer to publish his/her medical records for all to see.
Whatever method is introduced, it is clear that, as voters, we need to know.