I going to infer that it is only with Internet explorer users, is that the same with you?
I do not like IE. I'm using FF browser & it do mess the data. All began with FF not saving stored passwords & nicknames data of uCoz websites. To be safe I use different passwords for each account & I do need this option because I have my static PC locked & no one else have access to it. After some time it didn't save more than only passwords. FF didn't save the 'sections' which I read in uCoz forums already, even after I have marked them. I tried to re-install FF but it didn't work. I cannot tell you where is the problem. I have re-installed my OS & this problem is gone for me.
But I do believe your friends got some kinda of other problem(s). For example, they do not mark 'remember' & so on. Your post proves this problem got nothing to do with uCoz as you have tried to login with their user-names & everything worked fine, which means this thread got nothing to discuss about.