In theory they could have a problem with really sick people moving to California for the healthcare, raising their costs. But that takes time, so maybe before it becomes a big problem, other states will do similar things.
This isn't really an "in theory." Welfare states are fundamentally incompatible with open borders until we live in a world where equivalent levels of welfare are available everywhere. The incentives just don't add up.
On the other hand it could still work if the benefit is only available to Californians who have lived there for X years. Not sure about the legality of this though -- states can discriminate this way for in-state University tuition, but in Zobel v. Williams the Supreme Court ruled that Alaska couldn't pay out Permanent Fund dividends based on how long you had lived in Alaska (https://en.wikipedia.org/wiki/Alaska_Permanent_Fund).
In practice, I wonder if there'd be a grace period, e.g. your insurance doesn't kick in until you've lived in California a year. However, since so many jobs start your insurance benefits on day 1, I'd expect this would have to as well.
Of course, they also pay lots of taxes too, including sales tax, and presumably some of that, directly or indirectly, would offset the costs of this kind of program.
If that were true there would be no need for taxes. But in reality poor people will steal from non-poor people. That's what taxes are. "everybody pays" is a worthless statement if you don't include amounts.