My spouse is foreign and we both work and reside in a foreign country. I earn no income in the U.S. We were married this year and I am not sure what my filing status should be. My spouse does not want to apply for a SSN or Tax payer number as we have no intention to work or live in the U.S. Do his foreign earnings need to be reported? What should my filing status be?

If your spouse has no desire to get a number you will file as "Married Filing Seperately"

His earnings are not reported since he is a nonresident alien with no taxable US income.

If you are a US citizen you must file a US tax return no matter where you live and work converting your currency into dollars.

Citizens abroad have an automatic extension to file until June 15th. You may extend this to Oct. 15th by filing Form 4868 before June 15th.

You must file your tax return if your worldwide income exceeds the filing requirements. You can file as married filing jointly or married filing separately (MFS). On the MFS return, for your spouse SSN you can write "N R A".
You will also claim credit for taxes paid in the foreign country Form 1116 or/and for foreign earned income exclusion Form 2555.