I'm part of a team of 6 (1 senior (myself) 3 'standard' and 2 junior/trainee) that would broadly be described as "data analysts" and we report to a 'Analytics Manager'. Our everyday work isn't really important to the question but we are essentially tasked with routine reporting, answering ad-hoc questions from business people using the information we have access to , and attending meetings to present results.
We have a fair amount of turnover in this team as there is often a route from junior/standard analyst into a more technical role in another department.
My situation now
Our newest hire, 'Jane' has been with us for about 18 months. She does good work but in that time I have become increasingly annoyed and unsettled with her. The problem is she seems to see everything in terms of "certain doom"... any meeting that gets called will bring bad news, she snoops on shared meeting room calendars (and free-busy information on personal calendars) to try to find out if anything is going on. "Who is in the same meetings? Sam from HR is showing as busy at the same time as our boss and Department Manager Hannah!" etc.)
Any time a meeting gets called with all of us Jane will comment something like "Oh, we're about to find out our fate...." She genuinely seems to be afraid or apprehensive about the meeting (not just psyching us out) as I have seen her numerous times shaking, unable to eat etc. ahead of what are actually routine, but last-minute meetings.
Even in the absence of meetings she will start speculating about the future of our team, like "I'm sure they are looking into ways it could be automated" or things like that. She's explicitly said she thinks our department has a max lifespan of 2-3 years so she's already planning what she'll do about being laid off. She gossips about meetings she's seen on the shared calendars like "'Resourcing review with John and Claire?' They must be looking to spend less on the XYZ project?" ...
Her most recent days off she said she was preparing her house for (selling) downsizing and so on.
In general any email, announcement, development that happens... she seems to go into it with the nuclear sense that "well, that's it! We're definitely all going to be laid off" and then when it doesn't happen - she persists with all of us with questions like "do you think so-and-so was lying?" "What did you make of what X said, is it plausible?" "well, they didn't say anything at this meeting but how can we tell whether that's genuine or they were just lying again?" etc.
I was upbeat about my role, and about the future of the company, but I'm starting to wonder if Jane is on to something.
I may be a 'senior' analyst but I don't have any management insight as in privileged information or anything like that.
Question: How should I approach the Jane situation? Can I speak to her directly (and with what?) or should I take it to management for example?
 as an example of the sort of thing we do: A product manager might ask "What % of customers who took up a new subscription as part of a retention offer cancelled it within 12, 24 and 36 months and what reason did they give for cancelling?" We don't have a standard report for that, but we know how to extract that info.
None of the "ad hoc requests" have had anything to do with the future viability of the company, as far as I can tell.