No. This would cause confusion between your firm and the official Social Security Administration.
Social Security would not appreciate this and would probably demand that you change your company name.
If you feel that the name of your company is so distinctive that you want to protect it from use by others, then trademark your company name. The average cost to trademark a name is approximately $350 if you register it yourself.
Stop using their name. You should especially stop using its company name if the owner can provide evidence to support the claim of trademark infringement.
Yes, there are a few commonsense rules for choosing a company name?
1. Keep it short.
2. Make it relevant to the what the company offers.
3. Never use offensive language.
4. Make it memorable if possible.
5. Descript titles are effective.
6. Pick a company name carefully and try not to change it once marketed to the public.
7. Get some feedback about the name.
8. Look for a downside related to the name you like before making a final decision.