Is this possible (for a single-person corporation) in United States?
Yes, but if the salary you pay yourself is too low then you risk getting flagged for not providing yourself with reasonable compensation. A lot of resources recommend you that you pay yourself partially in dividends to avoid some taxes, as you mentioned is possible in the UK. However if you report no income you will probably be flagged by the IRS triggering an inquiry, audit, and potentially owe back taxes.
If yes, what company form would it require?
This is possible by organization your business into an S Corporation. The linked IRS site contains the necessary forms and instructions to register as such.
What would be the tax consequences?
Generally, you pay employer taxes and employee taxes on your salary but only pay employee income taxes on distributions. Some states (such as California) put extra taxes on S Corporations which may reduce some savings, though Florida's page on S Corporations does not mention any additional taxes. There are also filing and accounting costs associated with registering as an S Corporation. This could cost you around $100.