I'll do my best to answer your question in a simple and straightforward way.
First, a glossary of terms:
Congressional Health Plan -- every candidate has used this term, and it's a bit misleading in my opinion. The real name for this plan is FEP, or Federal Employees Program. The FEP program is an administrative services only (ASO) that is contracted out to various health insurance companies throughout the country. It's usually administered by a Blue. It uses a Blue's network, but has it's own fee schedule (maximum amount it will pay a provider for a given service).
When I worked with BCBSGA, the FEP unit was the dregs. They paid their claims processors and customer service and managers less that the regular employees, they had the hand me down computers and copiers, and just generally felt like you'd stepped into the scene of the suicides in Beetlejuice. The only people who worked there were very new hires who immediately moved on to other departments if they were any good, or the people who were burnt out, bitter, and not particularly good at their jobs and were therefore willing to work for less.
When I worked with physicians and hospitals, the FEP patient was always the bain of our existence. Because the physician group/hospital is contracted with BCBS (*or whichever administrator) and they write into contracts that you will accept federal program patients, you must see them. However, you know that you will be paid only about 70% of your contracted BCBS* rate. It's also very likely that the claim will be processed incorrectly, and no amount of appealing will ever get it corrected. Just as likely, the claim will just be denied or multiple services will be bundled into one service (ie, a surgery will be bundled with the pre-op exam and instead of a $60 reimbursement and a $430 reimbursement, you'll be paid $60 and told to write off $430 because the surgery is part of the pre-op. The best practices learn to simply not bill for the pre-op, which means many services are never even billed. I believe that for this reason many of the cost savings calculated for such plans is very misleading.
This goes on because in a given market, the FEP carrier(s) are always the largest insurers. A physician or hospital cannot compete financially if they don't accept the insurance of a huge number of people. Also, the FEP population is generally a very small percentage of the patient base so it's not a huge burden to most providers. Further, I would venture to guess that members of Congress never experience problems like this. I'm willing to bet that members of Congress and their families, as well as other high-ranking Washington officials, have a special dedicated unit of the best and brightest working their claims and handling service.
Universal Healthcare -- The idea that healthcare is an unalienable right of American citizens, and that it is the responsibility of the federal government to provide healthcare to all Americans.
Single Payor Healthcare -- A subset of supporters of universal healthcare support a single payor (government) healthcare. Advocates of single payor healthcare believe everyone should be entitled to the same quality of healthcare, and that only way to achieve fairness in the system is move to a government controlled healthcare system that sets prices and removes insurance companies from the process. However, most SP proposals actually contract the services out as ASO to insurance companies, so they would still be a part of the process.
(I'm going to go ahead and post this in case we lose power -- it's storming.)