Earlier quoted context omitted.
There is no requirement for your employer to provide the HSA. Frequently the employer can get better deals on the HSA than you can individually (like no waived maintenance fees and the like) but you can for sure open your own. In fact once you have enough money in there that you are moving out of liquid investments then you definitely want to shop around for HSA much like you would for an investment account.
Can you actually fund a non-employer HSA with tax-deductible funds, or is that a benefit that only applies to payroll HSA contributions? If not, a lot of the benefit of an employer HSA disappears. (FWIW, I have an HDHP, employer HSA, and an individual HSA that I roll my employer funds to annually for lower-fee investing.)
If you have your HSA contributions made through your company's payroll department, then you don't have to pay FICA taxes on it.