I am sorry for your misfortune, but I believe this to be misdirected.
(1) Employers offering Healthcare options to employees is more of a failure of the healthcare system in the US than a positive employment perk.
(2) Your company chooses the Healthcare provider they partner with. You are likely ineligible to be covered by their provider.
(3) Contractors do not get benefits. This has nothing to do with you being international. If you were based in the US, you would not be provided with Healthcare either.
All contractors in every country all over the world do not get employee benefits and is more or less the definition of being a contractor.
Either you are mistranslating some words, or seem to be missing some valuable pieces of information that is making you conflate all these variables.
It does bother me quite a bit though that you signed a non-compete agreement, as you typically only see this with employees.
(1) This is not enforceable (assuming of course you aren't reusing their work for other clients)
(2) As a contractor you are running a business. You are responsible for taxes, paying your salary, insurance, and Healthcare.
(3) Another company cannot dictate how you run your business. You can grow your business, take on new clients, do whatever you want.
It sounds like you need to do either one of two things.
(1) Start running your contractor setup more like a proper contractor shop. Get advice from here about expanding your business.
or
(2) If you really want to be an employee of a US company you can ask if they will sponsor you for a visa.
Hopefully my response has not been too direct, but I do think you need to get out of this mindset of thinking you are an employee. You are a contractor, and should be able to leverage the benefits of that. Most notably the larger incomes contractors demand for the simple fact they have so much more overhead to run their business.
Businesses in turn are happy to pay inflated contractor costs because it means they do not have to provide them with benefits like insurance.
These are two different ways of working and should not be equating them as you seem to be.
If you really want to be an employee, pretty much the only options are asking them to sponsor you for a visa, or getting them to open a branch office in your country.
I hope this gives you some thoughts to push things in one direction or another, and I hope that things turn in your favor this year.
What country?
Sorry that happened to you. My family is dealing with multiple health issues too.
It might help if we know the country. Even US based contractors need to buy their own insurance. That's part of why contractors usually get a higher rate. $110k USD is likely to be a very large amount in your country, as it about 150% the median household income in the US.