I just had my Schengen visa appointment this morning in the UK and paid 300 pounds for my Schengen fees, Swisscare Schengen insurance, and roundtrip nonrefundable flight tickets with Swiss Air. However, as I am a exchange student from the U.S., my U.S. university just this afternoon(hours after my morning appointment) announced that all exchange students needed to return back to the U.S. immediately due to the recent coronavirus outbreak. I am wondering if I can get any refunds for what I have already paid?
You can cancel the insurance at any time before the coverage begins. The company website says
Before the beginning of coverage, ... you can request a full refund at any time.
Visa application fee
Your visa fee is probably lost, but you might try asking if you can adjust your application dates. I wouldn't hold my breath.
Subject to some conditions, you can probably rebook the air tickets without change fees:
SWISS will waive its rebooking fees for all newly booked flights worldwide with immediate effect and until March 31 and offer a one-off free rebooking. For existing bookings, the new waiver policy will apply worldwide to all flights booked by March 5 with a departure date up to April 30, 2020. Passengers will thus be able to rebook their flight ticket once to a new date up to December 31, 2020 without incurring any rebooking fee.
If you're lucky enough that your flight is cancelled, however, you might get a refund:
For cancelled flights: If you have booked through swiss.com or a Service Centre and you don't want to rebook your flight ticket, you can refund your ticket. To do so, please complete the refund form.
If you booked through an (online) travel agency, please contact the travel agency.
I say "might" because I don't know whether you'd be able to get a refund if the flight is cancelled after you rebook it.
Taxes and fees
Even if you cannot recover your airfare, you should be entitled to a refund of certain taxes and fees for flights that you didn't actually take.
Possible insurance claims
If you have any travel insurance or other general insurance, you may want to look at whether it would cover any losses you incur. Again, I wouldn't hold my breath, but it's better to check now rather than to find out after the claim deadline has passed that you might have been eligible for reimbursement. (The Swisscare Schengen insurance seems not to include trip cancellation coverage, but I did not look very closely.)