If your friend is suicidal then you need to drive him to the nearest A&E for assessment and referral to treatment and proper, qualified support.
If he is not suicidal then you need not have him to stay for 3 or 4 days, or take those days off work to get through his 'really low point' when this would impinge on your children's feeling of ease in their own home.
If he needs monitoring then he needs professional assessment, and you are letting your own ego needs trump common sense if you think you can help him through this rough patch.
You say your oldest has behavioural issues, and I suggest that you have a very short window to get a grip on whatever it is that is going on there before puberty kicks in and he literally grows too big for you to handle, ending up off the rails and in the sort of trouble young teen boys can end up in - the stuff that can seriously limit their life chances.
Take care of your own first, is what I am saying.
It's far too easy to indulge your ego by playing Good Samaritan with other people.
Focus instead on your child with behavioural issues. Get help for him.
I highly recommend family therapy for you and your DCs. There is stuff going on that needs to be sorted out.
www.aft.org.uk/view/index.html?tzcheck=1
Association for Family Therapy and Systemic Practice website here^
You need to work on your boundaries. It's very telling that your children complain that this man takes up too much of your time. Some people are emotional vampires. They are bottomless pits of need. Their friends either back off (because they have healthy boundaries) or they get sucked in to the extent that they offer a couch for 3 or 4 days to offer mental health support that they are not qualified to give regardless of the objections of their own children, and regardless of the fact that they are not qualified to offer the sort of monitoring you say this person needs.
Your children are not extensions of your personality. They are real, sentient beings, with needs that are your first priority in every situation. You need to identify those needs and especially the needs of the one with obvious issues, and do whatever it takes to sort that out. Sort out what is going on under your own roof first. You don't owe anyone else your time or energy.