Yes but not from the company.
You need to have paid 15 years worth of social security payments to be eligible for the pension. Your employer may have been paying this for the time you have worked with them, but not all employers do.
If your employer has not, you can pay yourself, it is not expensive. You can also pay make up payments. For example, if you have not paid at all, you can make lump sum payments.
I have done this in Shanghai, and so what I say next applies to Shanghai, but the rules and the sums involved may vary for Kunming.
You must have permanent residence status to apply.
The street committee has an admin office in each district that deals with all Social Security matters for residents in that district. The annual payment is 500 rmb. After 15 years payments (minimum) upon reaching the age of 60 you are eligible for a pension of 1600/m index linked. You can make up payments for missed years.
BIG CAVEAT. Once you have passed 60, you cannot make back payments. You will have to pay each year for 15 years before you are eligible to be paid your pension.
If you have permanent residence status you can also apply for a health card. This entitles you partial payments on medical expenses at state hospitals. I think it is 50% payment for out patient treatment and 70% for in patient treatments. Receipts must be presented to claim monies back. I think you get money off at pharmacies by presenting your health card at the time of purchase.
Currently most cards can only be used in city of issue, unless you apply to transfer your cover to the city you are resident in. However, the government is rolling out a national system and I believe the card can be used inter-city if both cities are on the national system (I stand to be corrected). Kunming connects to the national system in the next month or two.
NB Medical costs are not high in China. I have just had surgery on my knee ligament (torn completely and needed to be sewn back together). This was at a national level sports injury center.
It cost me under 30k rmb for all costs,
up to the date of discharge from hospital several weeks later. That was before any medical cost reclaim from social security.