pDuring a campaign stop in Oakland on Friday afternoon, Vermont Sen. Bernie Sanders said his “Medicare for All” plan would cover every American, including the swelling population of people in California who are struggling with homelessness, mental illness and drug addiction.

“When we talk about Medicare for All, we mean Medicare for all,” he said in response to a question. “Whether it’s cancer or mental illness, they are all health care needs that need to be addressed by Medicare for All.”

Sanders spoke at an event hosted by the National Nurses United, the country’s largest union of registered nurses. The union officially endorsed Sanders as the Democratic Party’s presidential nominee Friday — a major win for the senator as he slips in the polls behind his main progressive competitor, Massachusetts Sen. Elizabeth Warren.

Sanders and Warren agree on the broad contours of what free health care should look like in America — no premiums, no deductibles, no co-payments. But they differ in their visions of how to fund it and roll it out.

Even though the candidates are relatively similar, Martha Utz, 78, a retired nurse, was unwavering in her support for Sanders.

“I don’t trust her,” she said of Warren. “She has all these wonderful ideas, but no rhyme or reason on how to get them going.”

Candy Matzke, 58, who flew from Minneapolis for the event, said she also likes Warren — but said she connects more with Sanders’ vision for overhauling the entire health care system.

“Bernie’s energy and vision of revolution is what the country needs, and it will be most helpful to this world,” she said.

The Oakland event was part of Sanders’ swing through California, where he has nearly doubled his paid staff recently. The campaign also celebrated the opening of a new field office in San Francisco’s Mission District on Thursday — one of five offices in the state.

Sanders spent most of the Oakland event talking about his sweeping plan to expand health care to every American and listed some specifics on what the plan would cover, including dental, eye and home health care. And, he added, Medicare for All also means “dealing with health care issues like addiction, which should not be dealt with in jails.”

The senator also briefly mentioned his plan to end homelessness in America, which involves investing $1.48 trillion over 10 years in the National Affordable Housing Trust Fund to build, rehabilitate or preserve the 7.4 million affordable housing units he says are needed in the country.

“I am aware of the crisis that exists in California, and it exists all around the country,” he said. “You’ve got a trillion dollars in tax breaks for the 1% and large profitable corporations, and yet tonight we have half a million people sleeping out on our streets or in emergency shelters.”

Homelessness and housing has yet to become major talking points among the Democratic presidential candidates, even though they are a huge concern for many California voters. Cities like San Francisco have a growing homeless population, with many on the streets also struggling with mental illness and substance abuse.

The nurses union said said it supports Sanders because of his proposed nurse-to-patient staffing ratios, federal health care workplace violence protections, support for a strong Veterans Health Administration and plan for guaranteed health care.

“We are here today with a senator who has the same calling that nurses feel and an ally who has stood by the side of workers everywhere, whether it’s in the strike line or in the halls of Congress,” said Bonnie Castillo, the union’s executive director.

Trisha Thadani is a San Francisco Chronicle staff writer. Email: tthadani@sfchronicle.com Twitter: @TrishaThadani