To open or contribute to an HSA you must meet certain eligibility requirements. Specifically:

You must be covered by a qualified high deductible health plan on the first day of the month
You cannot be covered by any other health plan that is not a qualified high deductible health plan, including spouse’s health insurance
You cannot be covered [...]

Is an otherwise eligible individual who has access to free health care or health care at charges below fair market value from a clinic on their employer’s premises eligible to contribute to an HSA?
An individual is not prohibited from contributing to an HSA merely because the individual has access to free health care or health [...]

If your spouse is enrolled in a Health Care FSA, you are not eligible to contribute to a Health Savings Account. The FSA is available to spouse and tax dependents of the person with the FSA. It can be used for the same medical expenses as the HSA. As a result, you cannot contribute to [...]

Employees who are enrolled in the High Deductible Health Plan (HDHP) are eligible to contribute to an HSA.  The employee can not be enrolled in any other health plan coverage that provides medical benefits before the HDHP deductible is met, not claimed as dependent under someone else’s tax return, and not enrolled in Medicare.

Frequently Asked Questions – Health Savings Accounts

Contributions
Reimbursements
Taxes