No, you should not complete a Schedule C since it was not your intent to make money. In addition, since you received money/reimbursement for hosting the student, you are not entitled to the $50/month charitable contribution that is allowed for hosting some foreign exchange students.
You will need to enter the 1099-MISC by following the instructions in the FAQ below, however it will not be subject to self-employment tax (which it would be if reported on a schedule C) and will be reported as Other Income.
You will want to make sure that you select that "you did not intend to make money" and that it was "not like your normal job" when asked in the interview portion of TurboTax.
https://ttlc.intuit.com/replies/4770594
