President Trump said he would be open to having the government take equity stakes in companies that require federal help. But Democrats are pushing to direct more assistance to workers and families rather than to corporations.

In the meantime, as employers at global conglomerates and kitchen-table offices anxiously grapple with the economy’s partial shutdown, tens of thousands of laid-off workers like Mr. Blomer are jamming government websites and phone lines to apply for unemployment insurance.

In some states, overwhelmed systems collapsed under the weight.

“It was so frustrating,” said Tim Tilley, who was laid off from his kitchen job at an Olive Garden in Ohio on Tuesday. For four hours that day and eight hours on Wednesday, he tried to file a claim. The website crashed repeatedly — after three attempts, he was locked out. He called dozens of times but was bumped from prompt to prompt, only to end up at the original automated message.

Rules for unemployment benefits vary across the country. States follow federal guidelines but administer their own programs. Each state uses its own formula to decide what percentage of weekly wages will be covered (50 percent, for example) and for how long (generally 26 weeks).

Emergency legislation that Mr. Trump signed into law on Wednesday would increase funding for states with surges in claims and pare back eligibility restrictions like waiting times and job search requirements. Although the rules are shifting, policy experts said, workers who have been quarantined or furloughed without pay should qualify.

Still, many workers could be left out entirely. Some states do not cover part-time workers, and many others have made it difficult for temporary workers to qualify. Gig workers are also often ineligible because they are typically considered self-employed.

In Maryland, for example, many applicants are confused about whether they would qualify for benefits if they became sick from the coronavirus and had to take time off or quit. The answer: No. Self-quarantines don’t qualify, either. But if a business had to shut down or lay workers off because of the virus, they could qualify.