Modern manners would seem to dictate that maintaining eye contact
Jobseekers are taught to engage in positive eye contact with their interviewers; children are exhorted to look someone in the eye if they speak to them; blind-daters are lectured about the necessity of gazing at their opposite number; everyone agrees that catching a barman's eye is the best way to get served.
But as with most diktats, it's best not to go too far with eye contact. In a sauna, shower or other gym situation, eye contact with anything other than the wall or your own navel is construed at best as a come-on, at worst as an affront.
If you're so obsessed with maintaining eye contact in an interview or on a date that you actually forget to maintain the conversation, then any good will be undone.
Remember that there's a mere blink between gazing and staring. Staring is never good. To a drunk, the tiniest glance can seem like aggressive staring, so avoid eye contact in such situations.
Catching someone's eye on a train, in a bar or in the park can be interpreted as an invitation to conversation whether you like it or not, so prepare for the consequences.
Children choose to have eye contact only with those they trust - perhaps that isn't such a bad lesson for all of us.