A negative test is required before anyone can enter the U.S. However, some who have crossed the southern border have tested positive for COVID-19.
PolitiFact notes that Abbot's claims are exaggerated. According to NBC, the city of Brownsville (which is near the border to Mexico) has confirmed that — since they had started carrying out tests in January — there have been 108 confirmed cases of COVID-19 which comes to 6.3 percent of those tested. Brownsville apparently added that they do not have the right to retain migrants but recommend that they isolate and stay in quarantine.
The U.S. State Department has also said that they coordinate with the CDC, Mexican government, and international organization partners to fully comply with the federal, state, local health orders.
The COVID-19 pandemic has given rise to a lot of potentially dangerous misinformation. For reliable advice on COVID-19, including symptoms, prevention, and available treatment, please refer to the World Health Organization or your national healthcare authority.