One could give a skeptic's answer for any form of this question in the context of a political campaign: presumably the candidates feel this answer will be more likely to get them nominated by their party in their campaign for President of the United States.
However, although this position is not made explicit, it is consistent with the idea included in the 2016 Democratic Party platform that describes healthcare as a right.
Since you mention Joe Biden in particular, he has said in a recent interview:
“I think undocumented people need to have a means by which they can be covered when they’re sick,” he said in a CNN interview, adding, “This is just common decency.”
“In an emergency they should have health care. Everybody should,” he added. "How do you say 'You're undocumented, I'm gonna let you die, man?'"
text via The Hill, original interview at CNN.
Democratic party platform
The Democratic party platform from 2016 states (emphasis mine):
Democrats believe that health care is a right, not a privilege, and our health care system should put people before profits.
This could possibly be interpreted as health care being a right for American citizens, but it could also be interpreted as health care being a human right, and if illegal immigrants are humans then they would be deserving of that right. The latter interpretation is also consistent with the Democratic Party's position on immigration, which treats the current system as broken and acknowledges the role that undocumented immigrants play in the US economy:
Those immigrants already living in the United States, who are assets to their communities and contribute so much to our country, should be incorporated completely into our society through legal processes that give meaning to our national motto: E Pluribus Unum.
and specifically on healthcare:
We will work to ensure that all Americans—regardless of immigration status—have access to quality health care. That means expanding community health centers, allowing all families to buy into the Affordable Care Act exchanges, supporting states that open up their public health insurance programs to all persons, and finally enacting comprehensive immigration reform.
I would add that this platform position does not fit the promise of "free" health care in the title of this question, but that promise is not necessarily implied by the question asked of the candidates, either.
The New York Times has a recent article discussing the status quo for healthcare for illegal immigrants. Currently federal funds are not supposed to be spent for healthcare for undocumented immigrants, and they are ineligible at the federal level for Medicare and Medicaid, although there are some states that do provide coverage.
However, there currently is indirect federal funding for health care for undocumented immigrants through centers that treat everyone regardless of circumstance and indirectly through hospitals that are ethically and morally required to treat ill people, whose cost of care is absorbed by those who are insured. Essentially, people in the United States illegally who do not have insurance are in a similar situation to citizens who do not have insurance: the care they do have access to is emergency care rather than preventative care.
Biden's position relative to the status quo is not entirely clear, because if he limits his stance to emergency care, then emergency care already exists. I am unaware of and did not find specific proposals from any of the candidates for how they would change the status quo.
It seems like the field of Democrats running for President is moving towards treating healthcare as a human right. The candidates may also want to further separate themselves from US government policies that currently house immigrants, and particularly children, in poor conditions. At the moment, there are not specific policies proposed by the candidates to change the status quo, and their position is consistent with the Democratic Party platform from the last presidential election cycle.