Employers in the U.S. can fire anyone at any time for practically any reason, cutting off one's livelihood. That feels more likely than a government doing the equivalent of cutting off someone's social security. A company's goal is to maximize profits, not provide charity to people it no longer deems useful to its bottom line.
> Changing your employer is much simpler than changing your government.
Making changes at your place of employment is often impossible. We get direct chances to change our government every X years. And changing to a different employer often isn't that easy for many folks.
Changing your employment is individual and allows different offerings from different employers, matching the wants and needs of different people. Changing your government needs a majority consensus and at least some people will be in the minority who doesn't get what they want.
I don't think tying employment to healthcare is a good thing, but I don't think your argument gives a good reason, either.
Changing your employment is as simple as irritating your manager or missing some KPI, and getting fired in all these 'right to work' states. UBI would keep people who would otherwise live on the brink of utter chaos enough to sever their dependence on minimum wage slavery.
Your picture of people picking and choosing from enticing opportunities isn't remotely true for people at the bottom where this would help.
So fix your employment, minimum wage, etc. laws, too. UBI is a huge undertaking in terms of legal and social changes, we shouldn't consider it "either this change or no change".
Changing your government (at least in a democracy) is way easier than changing your employer, which are generally run with no input from employees (although some employee-run businesses do get to vote for their management).