"During he said that he felt that I was not good at my job, was unsupportive and that the other 10 members of my team all felt the same way. To say I was shocked is an understatement as I have never had this feedback and pride myself on having a good working relationship with all of my team members."
Look at the context. It doesn't seem very likely he's being truthful. More likely he is lashing out because he is not enjoying being managed.
"I left the meeting (there was another manager present) due to being really upset. I have spoken to another manager, not my line manager, who suggested I take the rest of the day off."
Why take the day off? I get that it wasn't nice to hear that, even from someone with a clear agenda. But why did you need the day off?
"My question is that, where do I go from here? Do I request a meeting with HR to discuss these allegations? What do I say? I have MH issues myself and this has really knocked my confidence."
If you want a meeting to discuss what he has said, it should be with your own line manager, not HR. They don't sound like 'allegations'. He's expressed an opinion (biased) that you are not good at your job, and are unsupportive. I wouldn't count either of those as allegations. I don't know how good you are at your job, but based on your post, you're definitely supportive. He's just pissed off!
"I'm really lost to know what to do next, I don't feel that I can return tomorrow and carry on. I've also really embarrassed myself by getting upset so now the whole department knows, as they could see I had been crying, that something has happened."
I understand you have MH issues, so I get that hearing something negative might be more upsetting for you than for others. But if I were talking to someone without MH issues, I'd say not feeling you can return and carry on because a pissed-off difficult and disgruntled member of staff you are managing was less-than-complimentary is a massive overreaction.
I get that's how you're feeling, but I think you need to hear the context that that feeling is not normal. 
I imagine sometimes it's difficult to know whether your reaction to something/someone is justified, or whether it is a result of your own MH issues.
I would suggest speaking to your line manager, explaining what has happened, and asking whether he/she has any concerns about how you are managing staff/whether any further complaints from anyone else normal have been raised. I'm sure the answers will be reassuring.
If you are really concerned about how other members of staff view you, you could meet with the rest of your team just in the course of normal one-to-ones, ask how things are, whether there is anything you can support them with, or whether they have any concerns.
Unfortunately as a manager, with 11 people reporting to you, and dealing with difficult staff members being managed more firmly because of a performance/conduct/attendance issue, you're going to get negative reaction sometimes. It's par for the course. You know how good your relationships are normally, don't let one disgruntled person undermine that.