Some companies have onerous restrictions on employees using their names in public, but hinting obliquely is fine. It's a matter of not appearing to endorse random, unvetted public posts by employees, not a matter of secrecy.
The OP says "Fortune 5". The candidates are Walmart, Exxon Mobil, Chevron, Berkshire Hathaway, and Apple. Which one of those has super bowl ads and uses Akamai? There's your answer.
No, I have my own hosting company. It is not related to the Fortune-5 I was speaking of. To clarify, I own an IT consulting company and I have been on contract with the fortune-5. I supply contract heads else where too - H1Bs.
I own my own company. In fact, I own two companies.
One of my customers is a fortune-5. I am contracted and I have a small team that I manage within.
If I must explain, I'm a serial entrepreneur. I've owned 3-4 companies in my past including a cable TV company too. It would be more accurate to say I have my hands in many things with not much time on my hands.
THe fortune-5 company is an automotive company genius.