If Congress is unable to reach an agreement regarding the budget on Friday, the government will close at midnight on Saturday April 9. From that time forward, only "essential" government workers remain on the job. Here is a breakdown by agency:
CBP: Inspection and law enforcement are considered "essential personnel" though they will probably have fewer officers on duty.

DOL: DOL is reportedly making contingency plans for a shutdown. The online systems, iCERT/PERM, may or may not continue to function. There will be no live support including any help lines and email responses.

DOS: In case of a govt. shutdown, DOS is expected to issue visas ONLY in life or death situations, quite literally. Otherwise, the consulates are likely not to function for visa issuance.

EOIR: EOIR will likely function only regarding its detained docket.

USCIS: Since many USCIS functions are funded by filing fees, we MAY see a much lesser effect on benefits and adjudications.
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FAQ FROM EMPLOYERS
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Q1. Our employees are deployed at government sites. Are we still obliged to pay the H-1 wage?
A1. Yes. In my view, that obligation continues unabated.

Q2. Can we ask such employees to use their paid leave?
A2. No. If the employees voluntarily wish to do so for their convenience, I feel that would be legal. But the employer may not force employees to do so.