I don't believe you can be denied a border entry or access to travel on account of your mental health. I is my understanding that you can be denied entry on the basis of whether your medication you have on your person is permitted in the country(ies) one wishes to enter. Obviously any troubles you have had with the law are going to arise with a basic background check. But these checks don't for example include such things as hospital stays, diagnoses, and so on. This is why it is vitally important to do your homework prior to your travel and ensure you have your signed prescription and/or official letter from your prescribing doctor on your person.

Research your medications prior to departure. Often the list of banned substances can be be found on government web pages outlining transport and travel for the nation you are travelling to. For example if you are travelling to Canada, the list is available on the Transport Canada web site.

It is entirely your responsibility to conduct this home work and take measures accordingly. For example, I traveled Central Europe last year and one of my medications was banned in Hungary.I did such research well in advance and as a result was able to have a medication change before I left. I traveled everywhere with a list of my medications, a prescription of the same, and a travel document completed by my doctor. Other than delays for a check on a computer for whether my medications met the criteria, I had no problem.Make sure too that you travel with your medications on your person and not in checked luggage.