Can a H-1/EAD worker claim unemployment benefits ?

No.

Definitely no for H1 and you can claim of EAD but, I think, there will be a very bad effect on GC process. www.immigration-law.com says that you might not even get your approval because you are becoming a public charge.
 
out of status

If you apply for unemployement money , you are just telling them that you are out of status. it will effect your employement based GC. If you are in family based GC, it is ok to do
 
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