A lot of Canadian and US based companies always are careful of their employees. Generally it is of no concern to them about what you do outside of work, but many larger companies and government based sectors, such as hospitals in this case, believe that you should be representing the company while on the job and outside of the job.
As I have had years of experience as a manager, lead, senior, etc. role, I too have had to have talks with people about similar situations like these. Sure I have more of an open mind about things because of the way I was raised and where I grew up, but I can't speak for any other manager out there.
Looking at the video as a person who doesn't know anything about airsoft, I can see how people may become concerned about your situation, because a lot of people I know don't know what the sound of a gun really sounds like. They probably are thinking it is real because of the suppressor and from all the movies they've watched I bet they think it's real.
The best thing to do is have a sit down discussion about this with your manager/director with HR present. Explain to them what airsoft is all about and what your video is displaying. Keep it up for them to reference, but be advised that you may have to take it down because you technically are a representative of the company you work for even off work hours (some companies are pretty relaxed about it). Sure you might have to go see someone about it, but you can simply explain it to the doctor what airsoft is and he'll probably have a good laugh about it because he won't have to say a word to you. He/she will be happy enough to get paid by your company and not have to even talk to you.
If all goes well you'll be fine and will pave the way for everyone else in the company. I myself work with four other people who are airsoft players and they paved the way for me so now I can have my airsoft guns shipped to work and my CEO just has a chuckle about it every time it comes in for me. Sure he is a little bit more cautious, but when you see a 50" hardshell rifle case in my office who wouldn't be.... but we have a lot of weapons in the office out in the open... bow and arrows, airsoft rifles and pistols, swords, daggers, and a bunch of other goodies. =)
... Just call me Blue ...