I have no problems with eye contact with animals.
But it really depends on the animal, the setting, & the context....Because eye contact is often apart of dominance & intimidation, I only give or receive eye contact with an animal if it feels proper to do so. I actually seem to interact and understand animals rather well, as far as picking up on and understanding their body language and other very subtle nuances that they convey.
I literally never talk to pets, or any animals, in words. I communicate with them solely through simple sounds (different clicks, whistles, etc), eye contact, facial expressions, body movements, and other intuitive "feels" . I just get on their level of understanding/communication, more simple, basic and "animal" like communication.
When it comes to people there's often just way too much coming at me to take it all in, process, understand, and then respond to properly.