Now is the time to read the contract that you probably signed up to when BUNAC offered you the job. Have you got any grounds to cancel under the terms of the contract eg ill health?
If the terms of your having to pay a further £1000, if this is clearly set out in your contract and you signed it, unless the terms are legally unfair, you are probably on the hook for it. The fact that you are a student and finnancially challenged is unfortunately for you not BUNAC's problem. If you haven't signed a contract, you would be in a better position to refuse to pay.
You could try your Student Union for some assistance, they may have had students in this situation before.
I take it that you've been placed? You must have signed the Loan & Guarantee form - "I, the applicant, understand that by completing the application form for the BUNAC "Summer Camp USA" or “KAMP” programme and by accepting a contract with a camp, I will have entered into two separate legally binding agreements. Specifically I understand and acknowledge that as a condition of the said agreements, BUNAC has lent or will lend me the full cost of my transatlantic travel (except surcharges and insurance), visa processing (except those fees imposed directly by the US Government or its agencies), orientation and other fees and services as indicated, in all $1640, and that the full amount of the said loan (less such proportion thereof as is recoverable by BUNAC as set out below) is repayable by me to BUNAC upon demand.
Normally, the full amount of the said loan is recoverable from the camp by BUNAC provided that I fulfil the terms of my contract to the satisfaction of the camp, which involves (inter alia) working at the camp for the full duration of the contract." etc etc