UI benefits is insurance based and is NOT reported to immigration. If you are not citizen, only work authorization is validated against INS (so h1b never qualifies for this), no other data are sent. Also , IRS is totally different entity. UI is handled by individual states which is overseen by DOL, not IRS. However, states and IRS share data for tax/collection/payment interception etc.

All states has some kind of interfaces to share claimants and their details in some way with other states/agencies. Though INS do not request any data ( so far), if they start requesting these details, it can be available almost immediately. For this , there should be some rule/law that requires state to share these details though.

if you are looking for change of status, despite the fact above, it is always a wise decision to discuss with your lawyer.