No. Health Savings Accounts don’t’ have a “use it or lose it” provision.
The money in your HSA is yours to keep, whether it’s money that you contributed yourself, or it’s money that your employer contributed on your behalf.
There is another type of employer-funded account, called a Flexible Spending Account (FSA), that does have a “use it or lose it” provision. But that’s not related to the HSA.