It is a matter of State Law versus Immigration Law. Many states (like New York or California) allow unemployment benefits even to foreign nationals, as long as they are authorized to work in the U.S.  

This may seem in direct conflict with USCIS regulations, which makes working a condition of valid status in the U.S. 
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Even though this is paradoxical, if it became necessary for someone in a temporary visa status (E, TN, L-1A or O1) to apply for unemployment benefits then he can do so [see more details below). 
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It is to be noted that for those in H-1B, the employer is in theory always subject to paying the prevailing wage (DOL's rule) and going on unemployment would be in contradiction of that so in case of change of salary, it would be recommended to amend the existing petition to reduce the number of hours and thus salary. 

Grace period: As an employee under a non-immigrant visa status, if your employment is terminated, you are allowed to stay a 60 day grace period in the United States, period during which you may seek sponsorship for an employment-based visa status with a different employer.
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People in the US pursuant to the Visa Waiver Program who are nearing the end of their 90-day stay may apply for a special extension of their stay called “satisfactory departure”. This satisfactory departure may be extended for 30 days. The main reason to be considered must be related to health issue due the COVID.

Unemployment is NOT considered to be a public benefit under the new public charge rule.

The rule explicitly lists benefits that can be considered in the public charge analysis, which are (Supplemental Nutrition Assistance Program (SNAP), Supplemental Security Income (SSI), Temporary Assistance to Needy Families (TANF), most forms of Medicaid, and certain housing assistance programs. Notably absent from this list are state-funded programs and unemployment insurance.
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Note also the USCIS's own website which states that services received by noncitizens related to COVID-19 should not negatively affect the public charge analysis:

• USCIS encourages all those, including aliens, with symptoms that resemble Coronavirus 2019 (COVID-19) (fever, cough, shortness of breath) to seek necessary medical treatment or preventive services.  Such treatment or preventive services will not negatively affect any alien as part of a future Public Charge analysis.  
  
• USCIS will neither consider testing, treatment, nor preventative care (including vaccines, if a vaccine becomes available) related to COVID-19 as part of a public charge inadmissibility determination, nor as related to the public benefit condition applicable to certain nonimmigrants seeking an extension of stay or change of status, even if such treatment is provided or paid for by one or more public benefits, as defined in the rule (e.g. federally funded Medicaid).
  
• USCIS considers the receipt of public benefits as only one consideration among a number of factors and considerations in the totality of the alien’s circumstances over a period of time with no single factor being outcome determinative.