@dm8167,
You have no relationship whatsoever with this dude, other than the fact that you work together. I'm sure he won't notice if you ignore him or if you pay attention, sorry to say. Might as well do what feels best for you.
And my suggestion is to have a life doing other things and not thinking about this guy. Find a boyfriend elsewhere (work relationships are a poor idea no matter what) and this guy and his staring or not staring won't matter one iota to you.