It sounds like from a professional point of view, it would be wise for you to attend. However, I wouldn't think it's compulsory to actually eat anything. You could toy with a fork and a cup of hot water if you felt awkward with empty hands.
Bringing your own food... Really depends on the situation. I assume you haven't seen anyone else do it. If it's a big gathering and you could unobtrusively eat your food and kind of make it look like you were eating what was provided, it would probably be okay. If it's a small gathering and you would be eating something very fragrant or attention-getting somehow, that could be awkward. I think you would have to be prepared to apologetically blame health concerns or something, like you were on a specific diet for medical reasons. It would take some delicacy because I think you don't want to imply that the provided food is bad--"it's not the food, it's me." If the morning tea is a casual, come-and-go gathering, would it be possible to eat first and arrive late, or to leave early and eat then?
I don't think it's exactly rude for Boss to provide a limited menu, especially as it sounds like this is supposed to be a perk. Though, if it's a weekly thing, more people than just you would probably appreciate a little more variety in the offerings--same amount spent each week, but maybe bagels or fruit/veggies every once in a while instead of sweets. I'm not sure how one would go about suggesting that, though--I think it would also take some delicacy.