I would agree that the school staff member is probably suggesting an autistic spectrum disorder.
I would also agree that this behavior sounds a little bit like social referencing, which is generally typical of younger children, but isn't out of the range for a latency-age kid who has already been identified as in need of a little additional coaching in social skills. In particular, he is probably quite aware that his read on other children's interests and expectations is not always accurate, so of course he's going to check in with the designated adult authority on social interactions. Adults are also more likely to scaffold social interactions, which means that looking at her increases the likelihood of him experiencing a positive outcome, which reinforces the behavior. Especially in a group of kids enriched for others who need social skills support.
Anecdotally, I find that this happens to me quite often in social situations. If I am present in a group of multiple people, the speaker generally makes eye contact with me more than with anyone else in the group. As I've been observing this phenomenon for decades, I've concluded that it's a response to the high degree of scaffolding and positive feedback that I provide as a listener. Individuals with less social confidence tend to hone in on me more than those with more confidence do. You probably haven't observed the eye contact behavior at home because he is comfortable with that social communication environment. His home setting may also have more integrated environmental scaffolding for social skills, so that any little quirks aren't really felt.
As a listener, when I encounter speakers who restrict their eye contact to me, I do things like pointing my eye contact toward other members of the group, to drag the speaker's eye contact in that direction, deliberately including other members of the group with focused nonverbals or verbal responses that invite their participation, or even discreetly repositioning myself in the group so that the speaker's eyes will track across other members of the group.
Eye contact is an area for development in many individuals with social skills vulnerabilities, including the majority of such, who are not on the autistic spectrum.
