I can't offer comment about participating in both an FSA and HSA in the same year.
However, whether you have a HDHP through your own individual policy, or through your employer's insurance policy, you are free to use either your employer's HSA or open your own HSA (or even a combination of both, as long as your total combined contributions each year doesn't exceed the max allowed by law).
It is true that if your employer makes the contribution, you save the FICA tax on your contribution (and a few employers even offer their own HSA 'match' or contribution!). However, read the fine print of your employer's HSA account - some of these HSA accounts are worse than some bank checking fees, and monthly fees of up to $10/month are not uncommon! Also, some HSA accounts only pay paltry 0.10% APY, with limited investment choices. So even if you 'save' the FICA by using your employer's HSA account, the combination of monthly fees and poor investment choices can more than (negatively) offset better options elsewhere.
For instance, I've maintained my own individual HDHP for the past 11 years, and when HSAs first came out, I maxed out the annual contribution. Because my goal was to accumulate a stash, I looked for which HSA providers offered the best investment options. The bank I'm with (HSA Bank) lets you link to an HSA account with TD Ameritrade and buy a whole host of investments that you want (stocks, bonds, mutual funds, etc.). The only downfall is that if your monthly bank Savings Account balance is less than $3k, they nick you for $3/month. (I keep mine minimized so I can invest as much as possible) And, if you maintain a TD Ameritrade account, they ding you for another $2/month for having that privilege. But, after several years of saving, my HSA balance income from preferred and common stocks dwarfs the fees, and makes it worthwhile. (the funny thing is, my HSA investment account has had (relatively) tremendous gains over the past 7 years, while ALL of my other investment accounts have had mediocre performances! Figures that my least-accessible funds would have had the investment decisions resulting in the highest growth rates. lol )
If your goal w/ the HSA is to deposit money, get the tax deduction, and then use that money for your ongoing healthcare expenses, then look for the account w/ the lowest fees....but if you're looking to grow the stash, then don't focus only on the short-term fees and fail to see the long-term effect of the investment options.