Can the U.S government require by law that employers and employees sign a business contract in a private business?

Arguably the government could under the Commerce Clause: that clause gives the federal government the power to regulate any business or commercial activity that affects "interstate commerce." Interstate commerce is any economic activity that crosses state lines. Nowadays, almost everything is interstate commerce. Have a commercial website? The internet crosses state lines. Use telephones in your business? Even local calls be routed across state lines, so often the use of phones is enough to render a business "interstate." Does your business use any inventory, any equipment, any parts, any supplies, any software, etc. that comes from outside your state? Then you are in interstate commercial. Is the postal service or Fed Ex or UPS in or for your business in any way? Those are part of interstarte commerce, and so therefore are you. Other than babysitting for local kids or watching your neighbor's dogs, it's hard to imagine a business that is not in interestate commerce.
The federal government uses its power over interstate commerce, via the Commerce Clause, to impose a minimum wage and require overtime (under the Fair Labor Standards Act); to bar certain kinds of discrimination in employment (that's what the EEOC does); to impose certain safety standards (e.g. OHSA); to set standards for when someon is an "employee" vs. an "independent contractor." Given that the federal government already regulates (wages, safety, discrimination), it is not a far reach to believe that a mandate for employment contracts would also be legal.