What are the requirements for re-entry into the United States as an US citizen with a passport?
Are you required to show proof of leaving the US, to begin with? If so what will happen if you lost or don't have any proof of leaving the US?
Are there differences between air entry vs land entry?
No. There is not such requirement because the U.S. is among a handful of countries that does not have exit immigration. Although they do get some data, it is not complete and therefore proof of exit may not exist in some cases.
This is particularly the case with land borders. I have crossed the US-Canada and US-Mexico borders several times. When crossing into Canada, you go directly to Canadian immigration. They share some data with U.S. immigration but not with the person crossing, so you would not get proof of exist. Going to Mexico, you may not even go through Mexican immigration, only cross the yellow line delimiting the US and Mexico. When I crossed the border at Tijuana, there was only an official standing there that said Welcome to Mexico and that was it. No passport checks, let alone an exist stamp.
The requirement for a US citizen entering the United States is to present a US passport. This passport establishes your citizenship, and the immigration department is required to admit US citizens if they can prove their citizenship. (There are some exceptions, see Western Hemisphere Travel Initiative.)
US citizens are not required to show proof of leaving. A US citizen may leave the US at any time without telling anybody.
There are some differences between air and land entry, see the WHTI link above.