Here are my observations from a networking perspective.
I suspect the intermittent behavior is due to the fact that there are many servers associated with eas.outlook.com, outlook.office365.com, and m.outlook.com (which are all basically the same thing). My guess is some servers are having issues, others aren't. Your phone/computer/etc will cache a DNS entry for a certain amount of time, then look it up again, at which point it gets a non-working server and you're down until that timeout expires and it tries again (and happens to get a working one)
Basically it appears the DNS hierarchy is
Main hostnames (which are all just aliases/cnames):
eas.outlook.com
outlook.office365.com
m.outlook.com
outlook.ms-acdc.office.com
www.outlook.com
Looking up any of these will return one of two servers (based on the timing and DNS server):
ooc-g2.tm-4.office.com (strangely sometimes this comes back as a cname/alias for the one below, other times it comes back as its own A record)
MNZ-efz.ms-acdc.office.com
Each of those two servers above has many IPs associated, for load balancing reasons. Doing multiple NSlookups over time or to different servers will return a different list of IPs each time.
What I can't tell without being at Microsoft is if those IPs are themselves load balancers with many servers sitting behind them (which may make any workaround impossible, unless those load balancers are doing SSL offloading and that is where the issue lies, in which case avoiding the problematic ones could still fix it).
For example right now I'm connected to 52.96.58.98 and that is working fine. Earlier I was connected to 52.96.35.178 and that was also working fine. 52.96.19.82 seemed to be fine but it changed before I could test it thoroughly. I just bounced my wifi and am now on 52.96.109.146 and that is working also.
One possible workaround
Put one of the working IPs in for server name. This will only work if you have the "accept all certificates" option, like I do on android. Otherwise it will have a certificate failure. There are some security concerns with doing this though.
The other is to edit your hosts file to point eas.outlook.com or whatever of the main servers above you're using to one of the working IPs, so it won't use DNS and will stick to that IP.
I haven't tested either one as currently I'm working fine, but if it goes down again I'll give it a shot. Again, if the IPs are just front end load balancers, it probably won't help,
It also won't help if the issue is on some authentication server sitting on the backend somewhere, which you can't control.
EDIT - Just go bounced to 52.96.109.226 and that seems to not be working. Switching to mobile data fixed it. Back on wifi and a new IP, and it is working again.
I can confirm putting an IP in the hostname and setting it to "accept all certificates" does work. I guess if it goes hard down again at some point I'll give that a try and see if it actually fixes it.
Edit jan 30 - sadly the workaround does not work, servers that were fine last night are giving the error now, oh well.