I have worked for a non-profit hospital for over ten years in Virginia. Do I lose my pension if I am terminated?

Probably not.  Most pensions are regulated by federal law (the Employee Retirement Income Security Act; ERISA).  Even where the employer makes all of the contributions, the employee becomes "vested" over a period of time.  How long it takes to become "vested" depends in part on the terms of the "Plan Documents" which create the pension plan.  In your case, you should be fully vested.  Therefore, if your employment ends you should not lose your pension.  However, that may not mean you can withdraw the funds immediately.  Again, your access to the funds in the plan depends both on federal law and the terms of the plan. You might want to ask your employer to give you the "Summary Plan Description" for the pension plan.  The SPD is a booklet which generally explains the terms of the plan, your rights, vesting, etc.  That is a good place to start in answering your questions.