Yes, you can have an HSA and make contributions to it even if you have VA benefits. Receiving preventive care services or treatment for a service-related disability from the VA does not disqualify an individual from participating in an HSA. Previously the law stated that aside from preventive care, veterans receiving VA services were ineligible to contribute to an HSA for the 3 months following the date of service. Effective January 1, 2016, the 3 month rule was amended, allowing individuals who have received treatment from the VA for a service-related disability not to be disqualified from contributing to an HSA. Recipients of preventive care remain eligible as well.
Internal Revenue Bulletin: 2004-33, Notice 2004-50, Q&A 5 (https://www.irs.gov/irb/2004-33_IRB/ar08.html) states the original 3 month rule:
Q-5. If an otherwise eligible individual under section 223(c)(1) is eligible for medical benefits through the Department of Veterans Affairs (VA), may he or she contribute to an HSA?
A-5. An otherwise eligible individual who is eligible to receive VA medical benefits, but who has not actually received such benefits during the preceding three months**, is an eligible individual under section 223(c)(1). An individual is not eligible to make HSA contributions for any month, however, if the individual has received medical benefits from the VA at any time during the previous three months**.
IRS Notice 2008-59, Q&A-9 (https://www.irs.gov/irb/2008-29_IRB/ar11.html) maintains that preventive services from the VA do not disqualify an individual, mentioning the 3 month rule:
Q-9. Is an individual an eligible individual if he or she is eligible for medical benefits through the Department of Veterans Affairs (VA) but only receives medical care that is disregarded coverage or preventive care from the VA and is otherwise an eligible individual?
A-9. Yes. Although an individual actually receiving medical benefits from the VA at any time in the previous three months** is generally not an eligible individual, this rule does not apply if the medical benefits consist solely of disregarded coverage or preventive care.
**H.R. 3236 - Surface Transportation and Veterans Health Care Choice Improvement Act of 2015, Title IV, Section 4007 (https://www.congress.gov/bill/114th-congress/house-bill/3236) allows veterans receiving VA services to contribute to HSA's regardless of the 3 month rule, effective January 1, 2016:
"(Sec. 4007) Amends the Internal Revenue Code to...treat a veteran receiving hospital care or medical services for a service-connected disability as not disqualified from participating in or contributing to a tax-preferred health savings account."